;/i»f 


IN  MEMORIAM 
FLORIAN  CAJORl 


•  Z'-    /^ 


THE 


ELEMENTS 


OF 


ANALYTICAL  GEOMETRY, 


By  ELIAS  LOOMIS,  LL.D., 

n 

PROFESSOR  OF  NATURAL  PHILOSOPHY  AND  ASTRONOMY  IN  YALE  COLLEGE, 
AND  AUTHOR  OF  *'a  COURSE  OF  MATHEMATICS." 


REVISED  EDITION. 


NEW    YORK: 

HARPER   &    BROTHERS,   PUBLISHERS, 

FRANKLIN     SQUARE. 

18  7  8. 


LOOMIS'S  SERIES  OF  TEXT-BOOKS. 


ELEMENTARY  ARITHMETIC.  166  pp.,  28  cents. 
TREATISE  ON  ARITHMETIC.  352  pp.,  88  cents. 
ELEMENTS  OF  ALGEBRA.    Revised  Edition.    281  pp.,  %\  05. 

Key  to  Elements  of  Algebra,  for  Use  of  Teachers.    128  pp.,  %\  05. 
TREATISE  ON  ALGEBRA.    Revised  Edition.    384  pp.,  %\  17. 

Key  to  Treatise  on  Algebra,  for  Use  of  Teachers.    219  pp.,  $1  17. 
ELEMENTS  OF  GEOMETRY.    Revised  Edition.    388  pp.,  $1  17. 

ELEMENTS  OF  TRIGONOMETRY,  SURVEYING,  AND  NAVIGATION.    194  pp.,  $1  17, 
TABLES  OF  LOGARITHMS.    150  pp.,  $1  17. 

The  Trigonometry  and  Tables,  bound  in  one  volume.    360  pp.,  |1  75. 
ELEMENTS  OF  ANALYTICAL  GEOMETRY.    Revised  Edition.    261  pp.,  |1  17. 
DIFFERENTIAL  AND  INTEGRAL  CALCULUS.    Revised  Edition.    309  pp.,  $1  17. 

The  Analytical  Geometry  and  Calculus,  bound  in  one  volume.    6T0  pp.,  $2  05. 
ELEMENTS  OF  NATURAL  PHILOSOPHY.    351  pp.,  ^I  25. 
ELEMENTS  OF  ASTRONOMY.    254  pp.,  |1  17. 
PRACTICAL  ASTRONOMY.    499  pp.,  |1  75. 
TREATISE  ON  ASTRONOMY.    351  pp.,  f  1  75. 
TREATISE  ON  METEOROLOGY.    308  pp.,  $1  75. 


Entered  according  to  Act  of  Congress,  in  the  year  1872,  by 

Harper    &    Brothers, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


PREFACE. 


The  stereotype  plates  of  my  Elements  of  Analytical  Geom 
etry  having  become  so  much  worn  by  long-continued  use  that 
it  was  thought  desirable  to  renew  them,  I  have  improved  the 
opportunity  to  make  a  thorough  revision  of  the  work.  In  do- 
ing this,  it  has  been  thought  best  to  extend  considerably  the 
plan  of  the  work,  and  accordingly  I  have  not  merely  added  a 
third  part  on  Geometry  of  three  dimensions,  but  have  intro- 
duced new  matter  in  nearly  every  section  of  the  book.  I  have 
aimed  to  illustrate  every  portion  of  the  subject,  as  far  as  prac- 
ticable, by  numerical  examples,  generally  of  the  simplest  kind, 
the  main  object  being  to  make  sure  that  the  student  under- 
stands the  meaning  of  the  formulae  which  he  has  learned.  In 
making  this  revision  I  have  been  favored  with  the  constant 
assistance  of  Prof.  H.  A.  Newton,  who  has  carefully  examined 
every  portion  of  the  volume,  and  to  whom  I  am  indebted  for 
numerous  suggestions  both  as  to  the  plan  and  execution  of  the 
work.  It  is  hoped  that  the  volume  in  its  present  form  will  be 
found  adapted  to  the  wants  of  mathematical  students  in  our 
colleges  and  higher  schools ;  and  that,  if  any  should  desire  to 
prosecute  this  subject  further,  they  will  find  this  volume  a  good 
introduction  to  larger  and  more  difficult  treatises. 


Digitized  by  tine  Internet  Archive 

in  2008  witin  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofanalytOOIoomrich 


CONTENTS. 


PART  I. 

DETERMINATE  GEOMETRY. 

SECTION  I. 

APPLICATION  OF  ALGEBEA  TO   GEOMETRY. 

P«ir« 

Geometrical  Magnitudes  represented  by  Algebraic  Symbols 9 

Demonstration  of  Theorems 10 

Solution  of  Problems 12 

SECTION  II. 
CONSTRUCTION   OF  ALGEBRAIC  EXPRESSIONS. 

Construction  of  the  Sum  or  Difference  of  two  Quantities .' 18 

Product  of  several  Quantities 19 

Fourth  Proportional  to  three  Quantities 20 

Mean  Proportional  between  two  Quantities 21 

Sum  or  Difference  of  two  Squares 21 

Roots  of  Equations  of  the  Second  Degree 22 

To  inscribe  a  Square  in  a  given  Triangle 26 

To  draw  a  Tangent  to  two  Circles 27 

To  divide  a  straight  Line  in  extreme  and  mean  Ratio 29 


PART  II. 

INDETERMINATE  GEOMETRY. 


SECTION  I. 

CO-ORDINATES  OP  A  POINT. 

Methods  of  denoting  the  position  of  a  Point. 32 

Abscissa  and  Ordinate  defined — Equations  of  a  Point 33 

Equations  of  a  Point  in  each  of  the  four  Angles 34 

Polar  Co-ordinates 35 

Distance  of  a  Point  from  the  Origin 38 

To  coavert  Rectangular  Co-ordinates  into  Polar  Co-ordinates 42 

SECTION  II. 

TUB  STRAIGHT  LINE. 

Equation  of  a  Straight  Line 44 

Pour  Positions  of  the  proposed  Line 46 

Equation  of  the  First  Degree  containing  two  Variables 49 

Equation  to  a  Straight  Line  passing  through  a  given  Point 51 

Equation  to  a  Straight  Line  passing  through  two  given  Points 52 

Angle  included  between  two  Straight  Lines 54 


VI  CONTENTS. 

Page 

Condition  of  Pei-pendicularity 5G 

Equation  to  a  Straight  Line  referred  to  Oblique  Axes 69 

Perpendiculars  from  the  Vertices  of  a  Triangle  to  the  opposite  Sides 61 

SECTION  III. 

TRANSFORMATION   OF   CO-ORDINATES. 

To  change  the  Origin  without  altering  the  Direction  of  the  Axes 64 

To  change  the  Direction  of  the  Axes  without  changing  the  Origin 65 

To  transform  an  Equation  from  Rectangular  to  Oblique  Co-ordinates 66 

To  transform  an  Equation  from  Rectangular  to  Polar  Co-ordinates 67 

SECTION  IV. 

THE     CIRCLE. 

Equation  to  a  Circle  when  the  Origin  is  at  the  Centre 69 

Equation  to  a  Circle  when  the  Origin  is  on  the  Circumference 71 

Equation  to  a  Circle  referred  to  any  Rectangular  Axes 72 

Polar  Equation  to  a  Circle 74 

Equation  to  the  Tangent  at  any  Point  of  a  Circle 76 

Equation  to  the  Normal  at  any  Point  of  a  Circle 78 

Co-ordinates  of  the  Points  of  Intersection  of  two  Circumferences 80 

SECTION  V. 

THE    PARABOLA. 

Definitions — Curve  described  mechanically *. . .  84 

Equation  to  the  Parabola  referred  to  Rectangular  Axes 85 

Equation  to  the  Tangent  at  any  Point  of  a  Parabola 88 

Equation  to  the  NoiTnal  at  any  Point  of  a  Parabola 91 

Where  a  Tangent  to  the  Parabola  cuts  the  Axis 92 

Perpendicular  from  the  Focus  upon  a  Tangent 93 

Intersection  of  a  Circle  and  Parabola 94 

Equation  to  the  Parabola  referred  to  Oblique  Axes 95 

Polar  Equation  to  the  Parabola 99 

Area  of  a  Segment  of  a  Parabola 100 

SECTION  VI. 

THE    ELLIPSE. 

Definitions — Curve  described  mechanically 1 03 

Equation  to  the  Ellipse  referred  to  its  Axes 104 

Curve  traced  by  Points 106 

Equation  when  the  Origin  is  at  the  Vertex  of  the  Major  Axis 109 

Squares  of  two  Ordinates  as  Products  of  parts  of  Major  Axis 110 

Ordinates  of  the  Circumscribed  and  Inscribed  Circles Ill 

Equation  to  the  Tangent  at  any  Point  of  an  Ellipse Ill 

To  draw  a  Tangent  to  an  Ellipse  through  a  given  Point 113 

Equation  to  the  Normal  at  any  Point  of  an  Ellipse 115 

The  Normal  bisects  the  Angle  formed  by  two  Radius  Vectors. 116 

Every  Diameter  bisected  at  the  Centre 117 

Supplementary  Chords  parallel  to  a  Tangent  and  Diameter 120 

Points  of  Intersection  of  a  Circle  and  Ellipse 122 

Sum  of  Squares  of  two  Conjugate  Diameters 124 

Parallelogram  on  two  Conjugate  Diameters 125 

Equation  to  the  Ellipse  referred  to  a  Pair  of  Conjugate  Diameters 126 


CONTENTS.  Vll 

Page 

Squares  of  two  Ordinates  as  Products  of  parts  of  a  Diameter 127 

Polar  Equation  to  the  Ellipse 128 

Directrix  of  an  Ellipse 130 

Area  of  an  Ellipse 131 

SECTION  VII. 

THE    HYPERBOLA. 

Definitions-.-Curve  described  mechanically 133 

Equation  to  the  Hyperbola 1 34 

Curve  traced  by  Points 137 

Equation  when  the  Origin  is  at  the  Vertex  of  the  Transverse  Axis 141 

Squares  of  two  Ordinates  as  Products  of  parts  of  the  Transverse  Axis 141 

Equation  to  the  Tangent  at  any  Point  of  an  Hyperbola 142 

Equation  to  the  Normal  at  any  Point  of  an  Hyperbola 144 

The  Tangent  bisects  the  Angle  contained  by  two  Kadius  Vectors 145 

Every  Diameter  bisected  at  the  Centre 146 

Supplementary  Chords  parallel  to  a  Tangent  and  Diameter 148 

Properties  of  Conjugate  Diameters 150 

Difference  of  Squares  of  Conjugate  Diameters 152 

Equation  to  the  Hyperbola  referred  to  two  Conjugate  Diameters 153 

Squares  of  two  Ordinates  as  the  Kectangles  of  the  Segments  of  a  Diameter. .  155 

Polar  Equation  to  the  Hyperbola 155 

Directrix  of  an  Hyperbola 158 

Asymptotes  of  the  Hyperbola 160 

Tangents  through  the  Vertices  of  two  Conjugate  Diameters 162 

Equation  to  the  Hyperbola  referred  to  its  Asymptotes 165 

Equation  to  the  Conjugate  Hyperbola 166 

Equation  to  the  Tangent  referred  to  Asymptotes 168 

Intersection  of  Tangent  with  the  Axes 169 

SECTION  VIII. 

GENERAL  EQUATION  OP  THE  SECOND  DEGREE. 

The  Term  containing  the  first  Power  of  the  Variables  removed 170 

The  Term  containing  the  Product  of  the  Variables  removed 171 

Discussion  of  the  resulting  Equation 172 

Lines  represented  by  the  general  Equation  of  the  second  Degree 1 75 

Equation  to  the  Conic  Sections  referred  to  same  Axes  and  Origin 178 

SECTION  IX. 

LINES   OF   THE   THIRD   AND   HIGHER  ORDERS. 

General  Equation  of  the  Third  Degree 180 

Equations  of  the  Fourth  Degree 182 

SECTION  X. 

TRANSCENDENTAL   CURVES. 

Cycloid  defined 184 

Equation  of  the  Cycloid 1 85 

Logarithmic  Curve  defined 1 86 

Curve  of  Sines,  Tangents,  etc 187 

Spirals — Spiral  of  Archimedes — its  Equation 188 

Hyperbolic  Spiral — its  Equation 191 

Logarithmic  Spiral — its  Equation 192 


VIU  CONTENTS. 


PART  III 

GEOMETRY  OF  THREE  DIMENSIONS. 

SECTION  I. 

OF   POINTS   IN   SPACE. 

Pago 

Position  of  a  Point  in  Space  denoted 194 

Distance  of  any  Point  from  the  Origin 197 

SECTION  II. 

THE   STRAIGHT   LINE   IN   SPACE. 

Equation  to  a  Straight  Line  in  Space 199 

Equation  to  a  Straight  Line  passing  through  a  given  Point 201 

Equation  to  a  Straight  Line  parallel  to  a  given  Line 20 1 

SECTION  III. 

OF   THE   PLANE   IN   SPACE. 

Equation  to  a  Plane 206 

Equation  of  the  Plane  which  passes  through  three  given  Points 208 

Conditions  which  must  subsist  in  order  that  two  Planes  may  be  parallel 210 

Equation  of  a  Plane  perpendicular  to  a  given  Straight  Line 211 

SECTION  IV. 

OF   SURFACES   OF  REVOLUTION. 

Solid  of  Revolution  defined 214 

Equation  to  the  Surface  of  a  Right  Cylinder 214 

Equation  to  the  Surface  of  a  Right  Cone 215 

Equation  to  the  Surface  of  a  Prolate  Spheroid 216 

Equation  to  the  Surface  of  an  Oblate  Spheroid 217 

Equation  to  the  Surface  of  an  Hyperboloid 218 

Curve  which  results  from  Intersection  of  a  Cylinder  and  Plane. 220 

Curve  which  results  from  Intersection  of  a  Cone  and  Plane ? 221 

Curve  which  results  from  Intersection  of  a  Spheroid  and  Plane 227 

Curve  of  Intersection  of  a  Plane  and  Paraboloid 228 

Summary  of  preceding  Results 230 

SECTION  V. 

GENERAL   EQUATION  OF   THE   SECOND   DEGREE   BETVTEEN  THREE   VARIABLES. 

Classification  of  Surfaces  represented  by  the  general  Equation 233 

Particular  Cases  of  the  general  Equation 235 

Section  of  a  Surface  of  the  second  Degree  by  a  Plane 236 


APPENDIX. 

On  THE   GRAPHICAL  REPRESENTATION  OF  NATURAL  LaWS. 


ANALYTICAL   GEOMETRY. 


PART    I. 

DETERMINATE  GEOMETRY. 

SECTION  I. 
APPLICATION  OF  ALGEBRA  TO  GEO^IETRT. 

1.  We  have  seen  in  Geometry  (pages  40,  69,  and  162)  that 
all  geometrical  magnitudes,  including  angles,  lines,  surfaces, 
and  solids,  may  be  expressed  either  exactly  or  approximately 
by  numbers,  and  for  tliis  purpose  it  is  only  necessary  to  take 
one  of  these  magnitudes  as  the  unit  of  measure.  If  we  denote 
by  «,  5,  and  c  the  number  of  linear  units  contained  in  the  ad- 
jacent edges  of  a  rectangular  parallelopiped,  then  will  ah^  ac, 
ho  denote  the  magnitude  of  three  of  its  faces,  and  dbo  will  de- 
note its  volume. 

2.  In  like  manner,  every  geometrical  magnitude  may  be  rep- 
resented by  algebraic  symbols,  and  the  relations  between  dif- 
ferent magnitudes,  or  different  parts  of  the  same  figure,  may 
also  be  denoted  by  symbols.  AVe  may  then  operate  upon  these 
representatives  by  the  known  methods  of  Algebra,  and  thus 
deduce  relations  before  unknown ;  and  since  the  operations 
are  generally  very  much  abridged  by  the  use  of  algebraic  sym- 
bols, the  algebraic  method  has  many  advantages  over  the  geo- 
metrical. This  method  is  applicable  either  to  the  solution  of 
problems  or  to  the  demonstration  of  theorems. 

3.  Geometrical  problems  may  be  divided  into  two  classes : 
determinate  and  indeterminate.  Determinate  problems  are 
those  in  which  the  number  of  independent  equations  is  equal 
to  the  number  of  unknown  quantities,  and  therefore  the  un- 

A2 


10  ANALYTICAL   GEOMETRY. 

known  quantities  can  have  but  a  finite  number  of  values.  In- 
determinate problems  are  those  in  which  the  number  of  inde- 
pendent equations  is  less  than  the  number  of  unknown  quan- 
tities involved,  and  therefore  the  unknown  quantities  may  have 
an  infinite  number  of  values. 

4.  If  it  is  required  to  determine  the  magnitude  of  certain 
lines  from  the  knowledge  of  several  other  lines  connected  with 
the  former  in  the  same  figure,  we  first  draw  a  figure  which  rep- 
resents all  the  parts  of  the  problem,  both  those  which  are  given 
and  those  which  are  required  to  be  found.  We  denote  both 
the  known  and  unknown  parts  of  the  figure,  or  as  many  of 
them  as  may  be  necessary,  by  convenient  symbols.  We  then 
observe  the  relations  which  the  several  parts  of  the  figure  bear 
to  each  other,  from  which,  by  the  aid  of  the  proper  theorems 
in  Geometry,  we  derive  as  many  independent  equations  as 
there  are  unknown  quantities  employed.  By  solving  these 
equations  we  obtain  expressions  for  the  unknown  quantities  in 
terms  of  the  known  quantities. 

If  a  theorem  is  to  be  demonstrated,  we  express  by  algebraic 
equations  the  relations  which  exist  between  the  different  parts 
of  the  figure,  and  then  transform  these  equations  in  such  a 
manner  as  to  deduce  an  equation  which  expresses  the  theorem 
sought. 

5.  In  order  to  illustrate  these  principles,  let  it  be  required  to 
deduce  the  various  properties  of  a  right-angled  triangle  from 
the  principles  that  two  equiangular  triangles  have  their  ho- 
mologous sides  proportional,  and  that  tlie  perpendicular  drawn 
from  the  right  angle  of  a  right-angled  triangle  to  the  hypothe- 
nuse  divides  the  whole  triangle  into  similar  triangles. 

Let  the  triangle  ABC  be  right  angled  at  A ; 
from  A  draw  AD  perpendicular  to  BC,  and  let 
usputBC  =  a,AC=:J,AB  =  c,AD=A,BD=:m, 
^  and  DC  =  n.     Then,  by  similar  triangles,  we 
have  the  proportions 


APPLICATION   OF   ALGEBRA   TO   GEOMETRY.  11 

a:h::h:n  \  {h^=an,        (1) 

a\G\\c\m  \  whence  we  deduce \g^= am,       (2) 
a\G\\'b\h)  ( l)G=ah,        (3) 

Moreover,  we  have  a=:7n-\-n.         (4) 

These  four  equations  involve  the  various  properties  of  right- 
angled  triangles,  and  these  properties  may  be  deduced  by  suit- 
able transformations  of  these  equations. 

1st.  Equations  (1)  and  (2),  or  rather  the  proportions  from 
which  they  are  deduced,  show  that  each  side  about  the  right 
angle  is  a  mean  jprojpoTtional  between  the  entire  hyjpothenuse 
and  its  adjacent  segment, 

2d.  By  adding  equations  (1)  and  (2)  member  to  member,  we 
obtain 

V^^c^=am-\-an—a{rn-\-n^\ 
whence,  from  equation  (4),  we  obtain  lP'-\-c^=:a?\  that  is,  the 
square  of  the  hypothenuse  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides  of  the  triangle, 

3d.  By  multiplying  equations  (1)  and  (2)  member  by  mem- 
ber, we  obtain 

But  from  equation  (3)  we  have  also  U^c^^a^h^. 
Hence  a'^mn=:a'^h'^,  or  h^—mn;  that  is, 

m:h\:h\  n, 
OY^the  perj>endicular  drawn  from  the  vertex  of  the  right  aii- 
gle  ujpon  the  hypothenuse  is  a  mean  ^rojportional  between  the 
two  segments  of  the  hyjpothenuse. 

4th.  By  dividing  equation  (1)  by  equation  (2)  member  by 
member,  we  obtain 

^=^;  oYb'':c'\'.n:m; 
(?    am 

that  is,  the  squares  described  upon  the  sides  about  the  right 

angle  are  proportional  to  the  segments  of  the  hypothenuse. 

Thus  we  see  that  every  equation  deduced  from  tlie  equations 

(1),  (2),  (3),  and  (4),  when  translated  into  geometrical  languagCj 

is  a  geometrical  theorem. 


12  ANALYTICAL   GEOMETRY. 

6.  The  four  equations  of  the  preceding  article  contain  six 
quantities,  of  which,  when  a  certain  number  are  given,  it  may 
be  required  to  deduce  the  vaUies  of  the  other  quantities. 

Suppose  we  have  given  the  hypothenuse  BC,  and  the  per- 
pendicular AD,  and  it  is  required  to  determine  the  other  two 
sides  of  the  triangle,  as  also  the  two  segments  of  the  hypothe- 
nuse. 

We  have  already  found  H^-^c^^a^, 
and  fi'om  equation  (3)  we  have  21)G=2ah. 
By  adding  and  subtracting  successively,  we  obtain 

and  {h-cf  =  a?—2ah. 

whence  h-\-c=^a'^-\-'2ah;  h—c=^/d^  —  2ah. 

Knowing  the  sum  and  difference  of  the  two  sides  h  and  c,  by  a 

well-known  principle  (Alg.,  p.  89)  we  obtain 

the  greater  side  h=z^^a^-\-2ah-\-^^/a^—'^Mh^ 

the  less  side        c=^^/a'^-\-'^2,ah—^^/a^—''2ah. 

Siace  a^  J,  and  c  are  now  known  quantities,  the  two  segments 
are  given  by  equations  (1)  and  (2). 

The  preceding  principles  will  be  further  illustrated  by  the 
following  examples : 

Ex.  1.  The  base  and  sum  of  the  hyj^othenuse  and  perjpendic-- 
ular  of  a  right-angled  triangle  are  given,  to  find  the  perpendic- 
ular. 

Let  ABC  be  the  proposed  triangle,  right  angled 
at  B.  Kepresent  the  base  AB  by  J,  the  perpendic- 
ular BC  by  a?,  and  the  sum  of  the  hypothenuse  and 
perpendicular  by  s;  then  the  hypothenuse  will  be 
represented  by  s—x. 


A  B       By  Geom.,B.IY.,Pr.ll,  AB^  +  BC2=AC2; 

or,  J2  _|.  ^2  _  ^^  _  ^^2 — ^2  _  2sx  +  x^. 

Hence  h'^=s'^—28Xj 


^=-27 


that  is,  in  any  right-angled  triangle,  the  perpendicular  is  equal 
to  the  square  of  the  sum  of  the  hypothenuse  and  perpendicu- 


APPLICATION   OF  ALGEBRA   TO   GEOMETRY.  13 

lar,  diminished  by  the  square  of  the  base,  and  divided  by  twice 
the  sum  of  the  hypothenuse  and  perpendicular. 

Thus,  if  the  base  is  3  feet,  and  the  sum  of  the  hypothe- 

nuse  and  perpendicular  9  feet,  the  expression  —- —  becomes 
92—32  ^^ 

- — Q-=45  the  perpendicular. 

Ex.  2.  The  base  a?id  altitude  ofamj  triangle  are  given,  and 
it  is  required  to  find  the  side  of  the  inscribed  square. 

Let  ABC  represent  the  given  triangle, 
and  suppose  the  inscribed  square  DEFG 
to  be  drawn.  Eepresent  the  base  AB  by 
h,  the  perpendicular  CH  by  A,  and  the  side 
of  the  inscribed  square  by  x;  then  will 
CI  be  represented  by  h—x. 

Then,  because  GF  is  parallel  to  the  base  AB,  we  liave  by 
similar  triangles  (Geom.,  B.  lY.,  Pr.  16), 

AB:GF::CH:CI; 
that  is,  h\x\\h'. h—x, 

whence  hh—bx= hx  / 

hh 

that  is,  the  side  of  the  inscribed  square  is  equal  to  the  product 
of  the  base  and  height  divided  by  their  sum. 

Thus,  if  the  base  of  the  triangle  is  12  feet,  and  the  altitude 
6  feet,  the  side  of  the  inscribed  square  is  found  to  be  4  feet. 

Ex.  3.  The  base  and  altitude  of  any  triangle  are  given,  and 
it  is  required  to  inscribe  within  it  a  rectangle  whose  sides  shall 
have  to  each  other  a  given  ratio. 

Let  ABC  be  the  given  triangle,  and  sup- 
pose the  required  rectangle  to  be  inscribed 
within  it.  Bepresent  the  base  AB  by  b, 
the  altitude  CH  by  A,  the  altitude  of  the 
rectangle  DG  by  x,  and  its  base  DE  by  y ;  a  d  h  e  B 
also  let  aj :  y : :  1 :  7^/  or  y—tix. 

Then,  because  the  triangle  CGF  is  similar  to  the  triangle 
CAB,  we  have 


14  ANALYTICAL   GEOMETEY. 

AB:GF::CH:CI; 

that  is,  h:y\\h:h—x; 

whence  hh—hx=hy. 

But,  since  y=nx,  we  have 

hh—hx=hnx  ^ 

IK 

whence  x  =  7 7. 

o-\-nk 

If  we  suppose  n  equals  unity,  in  which  case  the  rectangle 
becomes  a  square,  the  preceding  result  becomes  identical  with 
that  in  Example  2. 

Ex.  4.  It  is  required  to  divide  a  straight  line  in  extreme 
and  mean  ratio  ;  that  is,  into  two  parts  such  that  07ie  of  them 
shall  he  a  mean  jprojportional  between  the  ichole  line  and  the 
other  part. 

Suppose  the  problem  to  be  solved,  and  that  C  is  such  a  point 

of  the  line  AB  that  we  have  the  proportion       . 

AB:AC::AC:CB.  ^  ^      ^ 

Put  AB = a,  AC = X,  whence  CB  =:a—x. 

The  preceding  proportion  wdll  then  become 
a\x\\x\a—x; 
whence  g?  —  d?-— ax, 

which  equation,  being  solved,  gives 

^'=-2-V^+4- 

Of  these  tw  o  values  obtained  by  the  solution  of  the  equation, 
tlie  first  is  the  only  one  which  satisfies  the  enunciation  of  the 
problem ;  for  the  second  is  numerically  greater  than  a,  and 
therefore  can  not  represent  z.jpart  of  the  given  line.  We  shall 
consider  hereafter  the  geometrical  signification  of  this  equa- 
tion. 

Ex.  5.  It  is  required  to  determine  the  side  of  an  equilateral 
triangle  described  about  a  circle  whose  diameter  is  given. 

Suppose  ABC  to  be  the  required  triangle  described  about  a 
circle  whose  diameter  is  given.  Draw  AE  perpendicular  to 
BC,  and  join  DC.  Eepresent  FE  by  d,  and  CE  by  x.  The  two 
triangles  ACE,CDE  are  similar,  for  each  contains  a  right,  an- 


APPLICATION   OF   ALGEBRA   TO   GEOMETRY.  15 

gle,  and  the  angle  CAE  is  equal  to  the  angle 

DCE.     Hence  we  have  the  proportion 

AC:EC::DC:DE. 

But  AC  is  double  of  EC ;  therefore  DC  is 

double  of  DE,  or  is  equal  to  d, 

Now  DC2-DE2=EC2, 

d? 
or  (^2_     _^2 

4 
whence  x—\d-\/Z^ 

or  2a?=c?-v/3; 

that  is,  the  side  of  the  circumscribed  triangle  is  equal  to  the 
diameter  of  the  circle  multiplied  by  the  square  root  of  3. 

Ex.  6.  Given  the  base  h  and  the  difference  d  between  the  hy- 
pothenuse  and  perpendicular  of  a  right-angled  triangle,  to  find 
the  perpendicular. 

Ans,  -^. 

Ex.  7.  Given  the  liypothennse  A  of  a  right-angled  triangle, 

and  the  ratio  of  the  base  to  the  perpendicular,  as  m  to  n^  to 

find  the  perpendicular. 

nh 
Ans,  — . 

Ex.  8.  Given  the  diagonal  6?  of  a  rectangle,  and  the  perime- 
ter 4^,  to  find  the  lengths  of  the  sides. 

Ans.j[>±.\J--p'. 

Ex.  9.  If  the  diagonal  of  a  rectangle  be  10  feet,  and  its  pe- 
rimeter 28  feet,  what  are  the  lengths  of  tlie  sides  ? 

Ans. 

Ex.  10.  From  any  point  within  an  equilateral  triangle,  per- 
pendiculars are  drawn  to  the  three  sides.  It  is  required  to  find 
the  sum,  s,  of  these  perpendiculars. 

Ans.  5  =  altitude  of  the  triangle. 

Ex.11.  Given  the  lengths  of  three  perpendiculars,  a,h,  and 


16  ANALYTICAL   GEOMETRY. 

c,  drawn  from  a  certain  point  in  an  equilateral  triangle  to  tlie 
three  sides,  to  find  the  length  of  the  three  sides. 

Jlns.         ^^      . 

Ex.  12.  Given  the  difference,  d,  between  the  diagonal  of  a 
square  and  one  of  its  sides,  to  find  the  length  of  the  sides. 

Ans.  d+d^/'2. 

Ex.  13.  In  a  right-angled  triangle,  the  lines  a  and  5,  drawn 
from  the  acute  angles  to  the  middle  of  the  opposite  sides,  are 
given,  to  find  the  lengths  of  the  sides. 


A71S.  2  Y  ,  and  2\J 


4:a''-b^ 


15     '  V       15 

Ex.  14.  In  a  right-angled  triangle,  having  given  the  hypothe- 
nuse  {a)y  and  the  difference  between  the  base  and  perpendicu- 
lar {2d),  to  determine  the  tv»-o  sides. 


Ans,  y  — ^ Yd,  and  y  — ^ d. 

Ex.  16.  Having  given  the  area  (c)  of  a  rectangle  inscribed 
in  a  triangle  whose  base  is  (J)  and  altitude  {a),  to  determine 
the  height  of  the  rectangle. 

.       a  ,      la?     ao 
^^'-  2*V  4-T- 
Ex.  16.  Having  given  the  ratio  of  the  tw^o  sides  of  a  triangle, 
as  m  to  n,  together  with  the  segments  of  the  base,  a  and  h, 
made  by  a  perpendicular  from  the  vertical  angle,  to  determine 
the  sides  of  the  triangle. 


Ans.  m\   —r- — :;,  and  n\    - 


m^—n"  V  m^'—n" 

Ex.  17.  Having  given  the  base  of  a  triangle  (2«),  the  sum  ol 
the  other  two  sides  (2«),  and  the  line  (c)  drawn  from  the  verti- 
cal angle  to  the  middle  of  the  base,  to  find  the  sides  of  the  tri- 


angle. 


Ans.  s±  ^Ja?j^G^—s^, 
Ex.  18.  Having  given  the  two  sides  {a)  and  {b)  about  the  ver- 
tical angle  of  a  triangle,  together  witli  the  line  (c)  bisecting 


APPLICAnON   OF   ALGEBEA   TO   GE03HETEY.  17 

that  angle  and  terminating  in  the  base,  to  find  the  segments  of 
the  base. 


Ans.  a\l — -J — ,  and  hsJ' 


ah—c^ 


ab    ^  ^     ah    ' 

Ex.  19.  The  sum  of  the  two  legs  of  a  right-angled  triangle  is 
5,  and  the  perpendicular  let  fall  from  the  right  angle  upon  the 
hypothenuse  is  a.    What  is  the  hypothenuse  of  the  triangle  ? 

Ans.  ^/s^-^c^^-a. 
Ex.  20.  Determine  the  radii  of  three  equal  circles,  described 
in  a  given  circle,  which  touch  eacli  other,  and  also  the  circum- 
ference of  the  given  circle  whose  radius  is  B. 

Ans,  K(2V3~3). 


18  ANALYTICAL  GEOMETEY. 


SECTION  11. 

CONSTEUCTION   OF  ALGEBEAIC   EXPEESSIONS. 

7.  The  construction  of  an  algebraic  expression  consists  in 
finding  a  geometrical  figure  which  may  be  considered  as  rep- 
resenting that  expression ;  that  is,  a  figure  in  which  the  parts 
shall  have  the  same  geometrical  relation  as  that  indicated  in 
the  algebraic  expression. 

The  elementary  exjpressions^  to  which  algebraic  expressions 
admitting  of  geometrical  construction  may  in  general  be  re- 
duced, are  the  following,  yiz. : 

_  _  ,  ^  ah 

x=a  —  o  +  c—a,etG,,    x=aOj  x=aoc,    os=—, 


x=Vad,  x=Va^-}-l>'^,  x=Va^^b'^; 

where  a,  5,  c,  etc.,  express  the  number  of  linear  units  contained 
in  the  given  lines. 

Problem  I.  To  construct  the  ex^pression  x^a+h. 

The  symbols  a  and  J,  being  supposed  to  stand  for  numerical 
quantities,  may  be  represented  by  lines.  The  length  of  a  line 
is  determined  by  comparing  it  with  some  known  standard,  as 
an  inch  or  a  foot.     If  the  line  AB  contains  the  standard  unit 

, L_     CL  times,  then  AB  may  be  taken  to  repre- 

^  sent  a.  So,  also,  if  BC  contains  the  stand- 
ard unit  h  times,  then  BC  may  be  taken  to  represent  h.  There- 
fore, in  order  to  construct  the  expression  a-\-l),  draw  an  indefi- 
nite line  AD.  From  the  point  A  lay  off  a  distance  AB  equal 
to  a^  and  from  B  lay  off  a  distance  BC  equal  to  h  /  then  AC  will 
be  a  right  line  representing  a-^h. 

Problem  II.  To  construct  the  expression  x—a—b. 

.  Draw  the  indefinite  line  AD.     From 

A.  c       B    D    ^^Q  point  A  lay  off  a  distance  AB  equal  to 

a,  and  from  B  lay  off  a  distance  BC,  in  the  direction  toward  A, 


CONSTRUCTION   OF   ALGEBRAIC   EXPRESSIONS.  19 

equal  to  5/  then  will  AC  be  the  difference  between  AB  and 
BC ;  consequently,  it  may  be  taken  to  represent  the  expression 
a—1). 

Problem  III.  To  construct  the  expression 
x=a—h-]-G—d+e. 

This  expression  may  be  written 

x=za-\-c-\-e—{J)-\-cl). 

To  obtain  an  expression  for  a-\-G-{-e^  draw  an  indefinite  line 
AX,  and  from  A  set  off  AB  —  a,     .       .   .  .     .         .      . 

from  B  set  off  BC=c,  from  C  set   ^     ^  =        ^   ""       "    ^ 
off  QT>  =  e;  then  A'D=a^-c+e. 

Then  set  off  from  D  toward  A,  DE=^/  from  E  set  off 
EF=^;  then  DF^J-f^. 

Hence  AF = fl^  4- c + 6  —  (J + 6?) = a?. 

In  a  similar  manner  we  may  construct  any  algebraic  expres-^ 
sion  consisting  of  a  series  of  letters  connected  together  by  the 
signs  +  and  — . 

In  like  manner  we  may  construct  the  expressions  x=3a, 
a; =55,  etc. 

Prohlem  lY.  To  construct  the  expression  x=ab. 

Let  ABDC  be  a  rectangle  of  which  the  side  AB  contains 
the  standard  unit  a  times,  and  the  side  AC  contains  the  same 
unit  h  times.     If  through  the  points  E, 
F,  etc.,  we  draw  lines  parallel  to  AC,        i — i — i — i — i — i 

and  through  the  points  G,  IT,  etc.,  we     h — K 

draw  lines  parallel  to  AB,  the  rectangle     G 1 

wild  be  divided  into  square  units.     In        ' — ^r—k — ^ — ' — 't. 
the  first  row,  AGIB,  there  are  a  square 
units ;  in  the  second  row,  GIIKI,  there  are  also  a  square  units ; 
and  there  are  as  many  rows  as  there  are  units  in  AC.     There- 
fore the  rectangle  ABDC  contains  axh  square  units,  or  the 
rectangle  may  be  considered  as  representing  the  expression  ab. 

An  algebraic  expression  of  two  dimensions  may  therefore 
be  represented  by  a  surface. 


20  ANALYTICAL   GEOMETRY. 

Problem  Y.  To  construct  the  expression  x=a^c. 

Let  there  be  a  rectangular  parallelopiped  whose  three  adja- 
cent edges  contain  the  standard  unit  respectively  a,  b,  and  c 
times ;  then,  dividing  the  solid  by  planes  parallel  to  its  sides, 
we  may  prove  that  the  number  of  solid  units  in  the  figure  is 
axbxc,  and  consequently  the  parallelopiped  may  be  consider- 
ed as  representing  the  expression  abc. 

An  algebraic  expression  of  three  dimensions  may  therefore 
be  represented  by  a  solid. 

Problem  YI.  To  construct  the  expression  x= — . 

From  this  equation  we  derive  the  proportion 
c:a::b:x; 
that  is,  a?  is  a  fourth  proportional  to  the  three  given  lines  c,  a, 
and  b. 

To  obtain  an  expression  for  x,  draw  two 
lines,  AB,  AC,  making  any  angle  with 
each  other.  From  A,  u23on  the  line  AB, 
lay  off  a  distance  AT>=c,  and  AB=ay 
and  upon  the  line  AC  lay  off  a  distance 
AE=b.  Join  DE,  and  through  B  draw  BC  parallel  to  DE; 
then  will  AC  be  equal  to  x. 

For,  by  similar  triangles,  we  have 

AD:AB::AE:AC, 
or  c:a::b:  AC. 

Hence  AC= — =x. 

c 

The  expression  x=—,  or  x= ,  may  be  co!^structed  in  the 

same  manner,  since  a?  is  a  fourth  proportional  to  the  three  lines 
c,  a,  and  a. 

Problem  YIl.  To  construct  the  expression  0?=-^-. 

This  expression  can.be  put  under  the  form 

ab    G 

X  =  -tX-. 

a     e 


CONSTKUCTION   OF  ALGEBRAIC   EXPRESSIONS.  3 

First  find  a  fourth  proportional  m  to  the  three  quar-ele 

d^  a,  and  h,  as  in  Prob.YI.     This  gives  ns  ^^^-r-     Tht^^e 

one  C : 

posed  expression  then  becomes  — ,  which  may  be  constru 

in  a  similar  manner. 

In  like  manner  more  complicated  expressions  may  be  con- 

structed ;  as   .^ /.^  . 

Problem  YIII.  To  construct  the  expression  x—  Vah. 

The  expression  Vab  denotes  a  mean  proportional  between 
a  and  h  /  for  we  have 

x^  =  axb ,'  OY  a'.x'.\x:b. 

To  construct  this  expression,  draw  an  in-  p 

definite  straight  line,  and  upon  it  set  off 
AB  =  a,  and  BC  =  b.  On  AC  as  a  diameter, 
describe  a  semicircumference,  and  from  B 


draw  BD  perpendicular  to  AC,  meeting  the  ^       ^ 

circumference  in  D ;  then  BD  is  a  mean  proportional  between 
AB  and  BC  (Geom.,  Bk.  lY.,  Prop.  23,  Cor.).  Hence  BD  is  a 
line  representing  the  expression  Vab=x. 

Problem  IX.  To  construct  the  expression  x—  Vct^-\-b'^. 

This  expression  represents  the  hypothenuse  of  a  right-angled 
triangle,  of  which  a  and  b  are  the  two  sides 
about  the  right  angle. 

Draw  the  line  AB,  and  make  it  equal  to  a; 
from  B  draw  BC  perpendicular  to  AB,  and 
make  it  equal  to  b.     Join  AC,  and  it  will  rep- 
resent the  value  of  Vd^  +  b'^\  since  AC^zrABHBC^  (Geom., 
Bk.IY.,Prop.ll). 

Problem.  X.  To  construct  the  expression  x=  Vd^  —  b^. 

This  expression  represents  one  of  the  sides  of  a  right-angled 
triangle,  of  which  a  represents  the  hypothenuse,  and  b  the  re- 
maining side. 

Draw  an  indefinite  hne  AB ;  at  B  draw  BC  perpendicular 
to  AB,  and  make  it  equal  to  b.     With  C  as  a  centre,  and  a 


20 


ANALYTICAL   GEOMETRY. 

radius  equal  to  a,  describe  an  arc  of  a  circle 

cutting  AB  in  D ;  then  will  BD  represent 

the  expression  Va^—b'^.    For 

BD2=DC2-BC^2_52. 

Whence  BD^Va'-b'^^x, 


Problem  XI.  To  construct  the  expression  x=Va^+b'^—c\ 

Put  a'^-\-b'^=zd'^,  and  construct  d  as  in  Prob.  IX.;  then  we 
shall  have  x  —  Vd"^ — c^^ 

which  may  be  constructed  as  in  Prob.  X. 

In  the  same  manner  we  may  construct  the  expression 
x—^/  a' —h^  -\-c^—d^  ^e^ — ,^iQ,. 

By  methods  similar  to  the  preceding  the  following  expres- 
sions may  be  constructed : 


1.  x—^/c^■\■ab.  4.  x=Vd^—bc, 

2.  x—'\/ab-\-cd.  5.  x=za'^-\-ab, 

lobe  o? 


^C 


Problem  XII.  To  construct  the  roots  of  the  four  forrns  of 
equations  of  the  second  degree  (Alg.,  Art.  277). 

In  the  equation  x^^jpx—  ±^, 

x^  and^a?  represent  surfaces  (Prob.  lY.) ;  q  must  therefore  rep- 
resent a  surface.  We  will  suppose  this  surface  transformed 
into  a  square  (^^),  and,  to  avoid  misapprehension,  will  write  the 
general  equation  of  the  second  degree 

QI?±JpX—±l^, 

First  form.  The  first  form  x^-\-jpx=l^  gives  for  x  the  two 
values     £c=— -^+V4+7i;2     and  a;=--|— Y^+^^. 

Draw  the  line  AB,  and  make  it  equal 
to  Ic.     From  B  draw  BC  perpendicular  to 

AB,  and  make  it  equal  to"^.     Join  A  and 
C ;  then,  as  in  Prob.  IX.,  AC  will  repre- 
sent the  value  of  \/"^4-^^ 


CONSTRUCTION   OF  ALGEBRAIC   EXPRESSIONS.  23 

With  C  as  a  centre,  and  CB  as  a  radius,  describe  a  circle 
cutting  AC  in  D,  and  AC  produced  in  E.  For  the  first  value 
of  X  the  radical  is  positive,  and  is  set  off  from  A  toward  C ; 

then  — ^  is  set  off  from  C  to  D,  and  AD,  which  equals 


^'■ 


?+*■-! 


represents  the  first  value  of  a?,  measured  from  A  to  D. 

For  the  second  value  of  x  we  begin  at  E,  and  set  off  EC 

equal  to  — -^ ;  we  then  set  off  the  minus  radical  from  C  to  A ; 

then  EA,  measured  from  E  to  A,  represents  the  second  value 
of  a?. 

Second  form.  The  second  form  x^—jpx^h^  gives  for  x  the 
two  values 

aj=|+\/^+P     and^=|-\/^+^. 

Construct  as  before  KQ  —  \J-j-\-h^\  then  from  C  lay  off  CE 
equal  to  "^j  and  the  fii-st  value  of  x  will  be  represented  by  AE, 
measured  from  A  to  E.     ^ 

From  D  lay  off  DC  equal  to"^;  then  from  C  in  a  contrary 

direction  lay  off  CA  equal  to  V^4r+^^ and  the  second  value 
of  X  will  be  represented  by  DA,  measured  from  D  to  A. 

Third  form.  The  third  form  d^-^-jpx^—T^  gives  for  x  the 
two  values 

Draw  an  indefinite  line  FA,  and  from 

7) 

any  point,  as  A,  set  off  a  distance  AB  =  — ■^. 

We  set  off  this  line  to  the  left,  because  "^  is  ^     ^^ ^E  A 


24  ANALYTICAL   GEOMETRY. 

negative.  At  B  draw  BC  perpendicular  to  FA,  and  make  it 
equal  to  h.  From  C  as  a  centre,  with  a  radius  equal  to  "^j  de- 
scribe an  arc  of  a  circle  cutting  the  line  FA  in  D  and  E.  Join 
CD,  and  we  shall  have  BD  or  BE  equal  to  sj'^—^^- 

The  first  value  of  x  will  be  represented  by  —  AB+BE,  which 
is  equal  to  —  AE.  The  second  value  of  x  will  be  represented 
by  —AB—BD, which  is  equal  to  —AD;  so  that  both  of  the 
roots  are  negative,  and  are  measured  from  A  toward  the  left. 

Fourth  form.  The  fourth  form  x^—jpx——lc'^  gives  for  x 
the  two  values 

x=\-V\l^-T^    and  a!=|-\/-f -y!;l 

Set  off  AB  equal  to  "^  from  A  toward  the 
right.  We  set  it  off  toward  the  right  be- 
^'  cause  w  is  positive.  Then  construct  the  rad- 
ical part  of  the  value  of  x  as  for  the  third  form.  To  AB  we 
add  BD,  which  gives  AD  for  the  first  value  of  x ;  and  from 
AB  we  subtract  BE,  which  gives  AE  for  the  second  value  of  x. 
Both  values  are  positive,  and  are  measured  from  A  toward  the 
right. 

Equal  roots.  If  the  radius  CE  be  taken  equal  to  CB,  that 

is,  if  Jc  is  equal  to  ^,  the  arc  described  with  the  centre  C  will 

not  cut  the  line  AF,  but  will  touch  it  at  the  point  B,  the  two 
points  D  and  E  will  unite,  the  radical  part  of  the  value  of  x  be- 
comes zero,  and  the  two  values  of  x  become  equal  to  each  other. 

Imaginary  roots.  If  the  radius  of  the  circle  described  with 

c  the  centre  C  be  taken  less  than  CB,  it  will  not 

meet  the  line  AF.    In  this  case  Jc^  is  numerically 

pi 
greater,  than  j-,  and  the  radical  part  of  the  value 

of  X  becomes  imaginary. 


CONSTRUCTION   OF  ALGEBRAIC   EXPRESSIONS.  25 

8.  Every  algebraic  expression  admitting  of  geometrical  con- 
struction must  have  all  its  terms  Jiomogeneous  (Alg.,  Art.  33) ; 
that  is,  each  term  must  be  of  the  same  degree.  The  degree  of 
any  monomial  expression  is  the  number  of  its  literal  factors. 
If,  however,  the  expression  have  a  literal  divisor,  its  degree  is 
the  number  of  literal  factors  in  the  numerator  diminished  by 
the  number  in   the   denominator.     Thus  the   expressions  x, 

ab  ahc      ',     ^    ^  ^  ^,  .         _  a^b  abed 

— ,  -7-'  are  or  the  nrst  degree ;  the  expressions  x^^,  — ,  -— r-  are 

of  the  second  degree.  In  order  that  an  algebraic  expression 
may  admit  of  geometrical  construction,  each  term  must  either 
be  of  the  first  degree,  and  so  represent  a  line ;  or,  secondly, 
each  must  be  of  the  second  degree,  and  so  represent  a  surface ; 
or,  thirdly,  each  must  be  of  the  third  degree,  and  denote  a  solid, 
since  dissimilar  geometrical  magnitudes  can  neither  be  added 
together  nor  subtracted  from  each  other. 

It  may,  however,  happen  that  an  expression  really  admitting 
of  geometrical  construction  appears  to  be  not  homogeneous; 
but  this  result  arises  from  the  circumstance  that  the  geomet- 
rical unit  of  length,  having  been  represented  in  the  calculation 
by  the  numeral  imit  1,  disappears  from  all  algebraic  expres- 
sions in  which  it  is  either  a  factor  or  a  divisor.  To  render 
these  results  homogeneous,  it  is  only  necessary  to  restore  this 
factor  or  divisor  which  represents  unit}^ 

Thus,  suppose  we  have  an  equation  of  the  form 

x=ab  +  c. 

If  we  put  I  to  represent  the  unit  of  measure  for  lines,  we  may 
change  it  into  the  homogeneous  equation 

lx—ab-\-€l, 

ab 
or  x=-j--\-c, 

which  is  easily  constructed  geometrically. 

Suppose  the  expression  to  be  constructed  to  be  of  the  form 


X: 


^-2c+3* 
B 


26  ANALYTICAL   GEOMETRY. 

Since  one  of  the  terms  of  the  numerator  is  of  the  second  de- 
gree, each  of  the  other  terms  of  the  numerator  should  be  made 
of  the  same  degree,  and  each  term  in  the  denominator  should 
be  made  of  the  first  degree ;  so  that,  introducing  the  linear 
unit  Z,  the  expression  to  be  constructed  is 

The  denominator  of  this  fraction  may  be  constructed  by  Prob. 
III.  If  we  represent  the  denominator  by  m,  the  expression 
may  be  written 

a"    Sib    2Z2 
m     m      m' 
each  of  which  terms  may  be  constructed  by  Prob.  YI. 

The  following  examples  will  show  how  an  algebraic  solution 
of  a  problem  may  be  converted  into  a  geometrical  solution. 

Problem  XIII.  Having  given  the  base  and  altitude  of  any 
triangle,  it  is  required  to  find  the  side  of  the  inscribed  square 
by  a  geometrical  construction. 

"We  have  found,  on  page  13,  the  side  of  the  inscribed  square 

to  be  equal  to  rZT/  that  is,  it  is  a  fourth  proportional  to 

h  -\-  A,  b  and  h. 

In  order  to  construct  this  expression,  produce  the  base  AB 
until  BL  is  equal  to  the  altitude  h;  through  L  draw  LM  par- 
allel to  BC,  meeting  CM  drawn  through 
C  parallel  to  AB.     Join  AM,  and  let  it 
meet  BC  in  P ;  draw  PE  perpendicular 
to  AB,  and  it  will  be  the  required  line. 
B    N        L  Draw  MN  perpendicular  to  AL. 
By  similar  triangles  we  have 

AL:AB::LM:Br::MISr:PE; 
that  is,  ^+A:^::A:rE; 

whence  PE — -j—-^ =w  : 

o-\-h       -' 

and  therefore  EP  is  equal  to  a  side  of  the  inscribed  square. 

Example  3,  page  13,  may  be  constructed  in  a  similar  manner 

by  laying  off  BL  equal  to  nh. 


CONSTKUCTION    OF  ALGEBRAIC   EXPRESSIONS. 


27 


•    Problem  XIY.  It  is  required  to  draw  a  straight  line  tan- 
gent to  two  given  circles  situated  in  the  same  plane. 

Since  the  two  circles  are  given  both  in  extent  and  posi- 
tion, we  know  their  radii  and  the  distance  between  their  cen- 
tres. 

Let  C,  C^  be  the  centres  of  the  two  circles,  CM,  CW  their 
radii.  Denote  the  radius  CM  of  the  first  circle  by  r,  that  of 
the  second  Q'W  by  ^'',  and  the  distance  between  their  centres 
CC  by  a.  Suppose  that  MM'  is  the  required  tangent ;  pro> 
duce  this  line  to  meet  QC  produced  in  T,  and  denote  the  dis- 
tance CT  by  a?. 

There  are  two  cases : 

Case  First.  When  the  tangent  does  not  pass  between  the 
circles. 

Draw  the  radii  CM,  CM'  to 
the  points  of  tangency ;  the  an- 
gles CMT,  C'M'T  will  be  right 
angles,  and  the  triangles  CMT, 
CMT  will  be  similar.  Hence 
we  shall  have  the  proportion 

CM:C'M'::CT:CT, 
or 
whence 

and 


r:r::x:  x—a 
rx—ra—r'x^ 
ar 


X—- 


.'  y 


from  which  we  see  that  CT  or  a?  is  a  fourth  proportional  to 
r—r\  a  and  r. 

To  obtain  a?  by  a  geometrical  construction,  through  the  cen- 
tres C  and  C  draw  any  tw^o 
parallel  radii  CN,  C'N',  on  the 
same  side  of  CC.  Through 
N  and  W  draw  the  line  XN"', 
and  produce  it  to  meet  CC 
produced  in  T.  CT  will  be 
the  line  represented  by  x. 

For  tlirough  W  draw  JST'D  parallel  to  CT;  then  ND  \viH 


28 


ANALYTICAL   GEOMETRY. 


represent  r— /•',  aud  WT>  will  be  equal  to  a  ;  and  by  similar 
triangles  we  have      B'^ :  T>W : :  CK :  CT, 
or  r—r^ :  a::r:  CT ; 

ar 


whence 


CTzz. 


Therefore  a  line  drawn  from  T,  tangent  to  one  of  the  cir- 
cles, will  also  be  tangent  to  the  other ;  and,  since  two  tangent 
lines  can  be  drawn  from  the  point  T,  we  see  that  this  first  case 
of  the  proposed  problem  admits  of  two  solutions. 

Oor.  If  we  suppose  the  radius  7'  of  the  first  circle  to  remain 
constant,  and  the  smaller  radius  r^  to  increase,  the  difference 
r— /  will  diminish ;  and,  since  the  numerator  ar  remains  con- 
stant, the  value  of  x  will  increase ;  which  shows  that  the  nearer 
the  two  circles  approach  to  equality,  the  more  distant  is  the 
point  of  intersection  of  the  tangent  line  with  the  line  joining 
the  centres.  When  the  two  radii  r  and  r^  become  equal,  the 
denominator  becomes  zero,  the  value  of  x  becomes  infinite,  and 
the  two  tangents  are  parallel. 

If  we  suppose  t*'  to  increase  so  as  to  become  greater  than  r, 
the  value  of  x  becomes  negative,  which  shows  that  the  point  T 
is  on  the  left  of  the  two  circles. 

Case  Second.  When  the  tangent  ^passes  between  the  circles. 

In  this  case,  as  in  the  other,  the 
lines  CM  and  Q'W  are  parallel; 
hence  the  triangles  CMT,  C'MT 
are  similar,  and  we  have  the  pro- 
portion 
C'M'::CT:CT,  '^''^^  '^^  '^'^ 
\r' \\x\a—x:         vL-^'-*--^)-  ^*" 


whence 


x= 


ar 

r-\-r' 


r-i  o- 


To  construct  this  expression,  through 
the  centres  C  and  C  draw  any  two 
parallel  radii  CN,  Q'W,  b'^^^g  ^^^  ^i^" 
ferent  sides  of  CC^;  join  the  jDoints 
NN^,  and  tln^ough  T,  where  this  line 


CONSTEUCTION    OF  ALGEBKAIC   EXPRESSIONS.  29 

intersects  CC^,  draw  a  line  tangent  to  one  of  tlie  circles.     It 
will  be  a  tangent  to  the  other. 

For  through  N'  draw  N'D  parallel  to  CC,  and  meeting  CN 
produced  in  D.  From  the  similar  triangles  NOT,  NDN'  we 
have  the  proportion 

]SrD:D]S['::NC:CT, 
or  r  i-r^  :a::r  :CT; 

whence  CT  =  — - — 7 = x. 

Cor.  The  value  of  x  is  positive  for  all  values  of  r  and  r' ; 

when  r=r'y  the  value  of  x  reduces  to  ^. 

If  each  circle  is  wholly  exterior  to  the  other,  there  may 
therefore  be  two  exterior  tangents  and  two  interior  tangents, 
in  which  case  the  problem  admits  offoitr  solutions. 

If  the  two  circles  touch  each  other  externally,  the  two  inte- 
rior tangents  unite  in  one,  and  the  problem  admits  but  three 
solutions. 

If  the  two  circles  cut  each  otlier,  the  interior  tangents  are 
impossible,  and  the  problem  admits  but  two  solutions. 

If  the. two  circles  touch  each  other  internally,  the  two  exte- 
rior tangents  unite  in  one,  and  the  problem  admits  but  one 
solution. 

If  one  circle  is  wholly  interior  to  the  other,  no  tangent  line 
can  be  drawn,  and  no  solution  of  the  problem  is  possible. 

The  general  values  of  x  already  found  undergo  changes  cor- 
responding to  the  changes  here  supposed  in  the  position  of  the 
two  circles. 

Prohlem  XY.  To  divide  a  straight  line  in  extreme  and 
mean  ratio. 

We  have  found,  in  Example  4,  page  14, 


x=-^±\/a''+- 


a" 

To  construct  the  first  value  of  x,  make  AB=a/  at  B  erect 
the  perpendicular  BC=^,  and  join  AC. 


30  ANALYTICAL    GEOMETRY. 

Then,  as  in  Prob.  9,  page  21, 


From  C  as  a  centre,  with  a 
radius  CB  =  ^,  describe  a  cir- 


cumference cutting  AC  in  D  and  AC  produced  in  E.     From 
AC  take  CD=^,  and  we  have 


AD:=AC-CDz=\/«2^-|'_| 


To  construct  the  second  value  of  x. 

From  E  set  off  EC  towards  the  left  equal  to  ^,  and  from  C 


v«'+? 


also  towards  the  left  set  off  CA  equal  to  \/  tj^^  +  -r.    Then  EA, 
measured  from  E  to  A,  will  represent 

^      <  /   2  ,  ^"^ 

With  A  as  a  centre,  and  AD  as  a  radius,  describe  the  arc 
DF.  The  line  AB  will  be  divided  in  the  required  ratio  at  F, 
and  AF  will  be  the  greater  part. 

The  second  value  of  a?=— AE  is  numerically  greater  than 
AB.  It  can  not,  then,  f  onn  a  part  of  AB,  and  is  not  an  an- 
swer to  the  question  in  the  form  here  proposed. 

Each  value  of  x  may,  however,  be  regarded  as  the  solution 
of  the  more  general  problem,  "  Two  points  A  and  B  being 
given,  to  find,  on  the  indefinite  line  that  passes  through  them, 
a  third  point  F,  such  that  the  distance  AF  shall  be  a  mean  pro- 
portional between  the  distances  AB  and  BF."  To  this  problem 
there  are  evidently  two  solutions,  F  on  the  right  of  A  being 
one  of  the  points,  and  F'  on  the  left  of  A  is  the  other. 

9.  From  the  preceding  examples  we  perceive  that  the  solu- 
tion of  a  geometrical  problem  by  the  aid  of  Algebra  consists 
of  three  principal  parts : 


CONSTRUCTION   OF  ALGEBEAIC   EXPRESSIONS.  31 

1^^.  To  translate  the  problem  into  algebraic  language,  or  to 
reduce  it  to  an  equation. 

26?.  To  solve  the  equation  or  equations. 

Sd.  To  construct  geometrically  the  algebraic  expressions  ob- 
tained. 

Frequently  it  becomes  necessary  to  add  a  fourth  part,  whose 
object  is  the  discussion  of  the  jprohlem^  or  an  examination  of 
all  the  circumstances  relating  to  it. 


32  ANALYTICAL    GEOMETRY. 


PART   II. 
INDETERMINATE  GEOMETRY. 

SECTION  I. 
CO-OEDINATES    OF   A   POINT. 

10.  The  object  of  the  second  branch  of  Analytical  Geometry 
is  to  determine  the  algebraic  equations  by  which  known  lines 
and  curves  may  be  represented,  and  from  these  equations  to 
deduce  their  geometrical  properties ;  and  conversely,  having 
given  the  equations,  to  determine  the  lines  and  curves  which 
they  represent. 

11.  To  determine  the  position  of  a  point  in  a  plane.  The 
position  of  a  point  in  a  plane  may  be  denoted  by  means  of  its 
distances  from  two  given  lines  which  intersect  one  another. 

Thus,  let  AX,  AY  be  two  assumed  straight 
lines  which  intersect  in  any  angle  at  A,  and 
let  P  be  any  point  in  the  same  plane  ;  then, 
if  we  draw  PB  parallel  to  AY,  and  PC  par- 
allel to  AJX,  the  position  of  the  point  P  will 
■X  be  determined  by  means  of  the  distances 
PB  and  PC. 


7 


The  two  lines  AX,  AY,  to  which  the  position  of  the  point  P 
is  referred,  are  called  axes^  and  their  point  of  intersection,  A,  is 
called  the  origin.  The  distance  AB,  or  its  equal  CP,  is  called 
the  abscissa  of  the  point  P ;  and  BP,  or  its  equal  AC,  is  called 
the  ordinate  of  the  same  point.  Hence  the  axis  AX  is  called 
the  axis  of  abscissas,  and  AY  is  called  the  axis  of  ordinates. 

The  abscissa  and  ordinate  of  a  point,  when  spoken  of  togeth- 
er, are  called  the  co-ordinates  of  the  point,  and  the  two  axes 
are  called  axes  of  co-ordinates,  or  co-ordinate  axes. 

A  system  of  axes  may  be  either  rectangular  or  oblique  ^ 
that  is,  the  angle  YAX  may  be  either  a  right  angle  or  an 


CO-OEDINATES    OF   A   POINT.  33 

oblique  angle.  Rectangular  axes  are  ordinarily  most  conven- 
ient, and  will  generally  be  employed  in  this  treatise. 

An  abscissa  is  usually  denoted  by  the  letter  a?,  and  an  ordi- 
nate by  the  letter  y  /  and  hence  the  axis  of  abscissas  is  often 
called  the  axis  of  a?,  and  the  axis  of  ordinates  the  axis  of  y. 

The  abscissa  qfanyjpoint  is  its  distance  from  the  axis  of 
ordinates  measured  on  a  line  parallel  to  the  axis  of  abscissas. 

The  ordinate  of  any  point  is  its  distance  from  the  axis  of 
abscissas  measured  on  a  line  parallel  to  the  axis  of  ordinates. 

12»  Equations  of  a  point.  The  position  of  a  point  may  be 
determined  when  its  co-ordinates  are  known.  For,  suppose  the 
abscissa  of  the  point  P  is  equal  to  5,  and  its  ordinate  is  equal 
to  4.  Then,  to  determine  the  position  of  the 
point  P,  from  the  origin  A  lay  off  on  the  axis 
of  abscissas  a  distance  AB  equal  to  5  units 
of  length,  and  through  B  draw  a  line  paral- 
lel to  the  axis  of  ordinates.  On  this  line  lay 
off  a  distance  BP  equal  to  4  units  of  length,  ^  „         ^^ 

and  P  will  be  the  point  required. 

So,  if  x—a  and  y=b^  measure  off  AB  equal  to  a  units,  and 
draw  BP  parallel  to  AY,  and  equal  to  b  units. 

Hence,  in  order  to  determine  the  position  of  a  point,  we  need 
only  have  the  two  equations 

x^a,y=b, 
in  wdiicli  a  and  b  are  given.     These  equations  are  therefore 
called  the  equations  of  a  point. 

13.  Bigns  of  the  co-ordinates.  It  is  however  necessary,  in 
order  to  determine  the  position  of  a  point,  that  not  only  the 
absolute  values  of  a  and  b  should  be  given,  but  also  the  signs  of 
these  quantities.  If  the  axes  are  produced  through  the  origin 
to  X'  and  Y',  it  is  obvious  that  the  abscissas  reckoned  in  the 
direction  AX'  ought  not  to  have  the  same  sign  as  those  reck- 
oned in  the  opposite  direction  AX,  nor  should  the  ordinates 
measured  in  the  direction  AY'  have  tlie  same  sign  as  those 

B2 


34:  ANALYTICAL   GEOMETEY. 

measured  in  the  opposite  direction 
AY ;  for  if  there  were  no  distinction 
in  this  respect,  the  position  of  a  point 
as  determined  by  its  equations  would 
be  ambiguous.  Thus  the  equations  of 
the  point  P  would  equally  belong  to 
the  points  P^  P^^,  V''\  provided  the 
absolute  lengths  of  the  co-ordinates  of  these  points  were  equal 
to  those  of  P.  This  ambiguity  is  avoided  by  regarding  the 
co-ordinates  which  are  measured  in  one  direction  Sisplus,  and 
those  in  the  opposite  direction  as  minus.  It  has  been  agreed 
to  regard  those  abscissas  which  fall  on  the  right  of  the  axis 
YAY'  as  positive,  and  hence  those  which  fall  on  the  left  must 
be  considered  negative.  So  also  it  has  been  agreed  to  consider 
those  ordinates  which  are  above  the  axis  XAX'  as  positive,  and 
hence  those  which  fall  below  it  must  be  considered  negative. 

14.  Eiiuations  of  a  jpoint  in  each  of  the  four  angles.  The 
angle  YAX  is  called  \hQ  first  angle;  YAX'  the  second  angle; 
Y'AX'  the  third  angle  ;  and  YAX  i\\Q  fourth  angle. 

The  following,  therefore,  are  the  equations  of  a  point  in  each 
of  the  four  angles  : 

For  the  point  P     in  the  first  angle,       x=-{-a,y—-\-'b. 

"  P'         "     second  angle,  a?  =—(3^,  2/=  rf  5. 

"  P'^       "     third  angle,     x=—a,y=—h. 

"  Y'[       "     fourth  angle,  x=  -\-a,  y^  —h. 

If  the  point  be  situated  on  the  axis  AX,  the  equation  y=ih 

becomes  ?/=0,  so  that  the  equations 

x=±a^y=0 
denote  a  point  in  the  axis  of  abscissas  at  the  distance  a  from 
the  origin. 

If  the  point  be  situated  on  the  axis  AY,  the  equation  x—a 
becormes  a?=0,  so  that  the  equations 

x  —  0,y—±h 
denote  a  point  on  the  axis  of  ordinates  at  the  distance  h  from 
the  origin. 


CO-OEDINATES   OF   A  POINT.  35 

•    If  the  point  be  common  to  both  axes,  that  is,  if  it  be  at  the 
origin,  its  position  will  be  denoted  by  the  equations 

jK^O,  2/=0. 

The  point  P,  whose  co-ordinates  are  x,  y^  is  often  called  the 
point  (ic,  y)\  thus  a  point  for  which  x—a^y—h  is  called  the 
point  («,  h).  Hitherto  the  letters  a  and  h  have  been  supposed 
to  stand  for  positive  numbers,  but  they  may  also  be  used  to 
represent  negative  numbers. 

Ex.  1.  Indicate  by  a  figure  the  position  of  the  point  w^liose 
equations  are  aj=  +4,  2/=  —  3. 

Ex.  2.  Indicate  by  a  figure  the  position  of  the  point  whose 
equations  are  a?==  —  2,  ?/=  -f  7. 

Ex.  3.  Indicate  by  a  figure  the  position  of  the  point  0,  —5, 

Ex.  4.  Indicate  by  a  figure  the  position  of  the  point  —  8,  0. 

Ex.  5.  Indicate  by  a  figure  the  position  of  the  point  —3,  —2. 

Ex.  6.  Draw  a  triangle  the  co-ordinates  of  whose  angular 
points  are  3, 4 ;  —3,-4;  —1,0. 

15.  Polar  co-ordinates.  The  position  of  a  point  may  also 
be  denoted  by  means  of  the  distance  and  direction  of  the  pro- 
posed point  from  a  given  point. 

Thus,  if  A  be  a  known  point,  and  AX  be  ,p 

a  known  direction,  the  position  of  the  point 
P  will  be  determined  when  we  know  the 
distance  AP,  and  the  angle  PAX.  . 

Thus,  if  we  denote  the  distance  AP  by  r^ 
and  the  angle  PAX  by  0,  the  position  of  P  is  determined  if  r 
and  0  are  known. 

The  assumed  point  A  is  called  the  pole  ;  the  distance  of  P 
from  A  is  called  the  radius  vector  ;  the  line  AX  is  called  the 
initial  line  /  and  the  radius  vector,  together  with  its  angle  of 
inclination  to  the  initial  line,  are  called  ihiQjpolar  co-ordinates 
of  the  point.  The  point  whose  polar  co-ordinates  are  r  and  0 
is  sometimes  called  the  point  r^  0.  f 


36  ANALYTICAL   GEOMETRY. 

16.  Unit  for  the  measure  of  angles.  The  unit  commonly 
employed  in  Trigonometry  for  measuring  angles  is  the  nine- 
tieth part  of  a  right  angle,  called  a  degree  •  but  a  different  unit 
is  sometimes  more  convenient.  Since  angles  at  the  centre  of 
a  circle  are  proportional  to  the  arcs  on  wliich  they  stand,  we 
may  employ  the  arc  to  measure  the  angle  which  it  subtends, 
and  it  is  convenient  to  take  as  the  unit  of  measure  the  arc 
which  is  equal  to  the  radius  of  the  circle.  Since  the  circum- 
ference of  a  circle  whose  radius  is  unity  is  27r,  the  measure  of 
four  right  angles  will  accordingly  be  27r ;  the  measure  of  one 

right  angle  will  be  -^ ;  the  measure  of  an  angle  of  45°  will  be 

J,  etc. 

17.  Negative  values  ofjpolar  co-ordinates.  The  position  of 
any  point  might  be  expressed  by  positive  values  of  the  polar 
co-ordinates  r  and  0,  since  there  is  here  no  ambiguity  corre- 
sponding to  that  arising  from  the  four  angles  of  the  figure  in 
Art.  13.    It  is,  however,  sometimes  convenient  to  admit  the  use 

of  negative  angles,  and  in  this  case  an  an- 
gle XAP'  is  considered  negative  when  it  is 
measured  in  the  direction  corresponding  to 
the  motion  of  the  hands  of  a  w^atch ;  and 
an  angle  is  considered  positive  when  it  is 

measured  in  the  opposite  direction,  as  XAP. 

The  same  direction  may  be  represented  either  by  a  negative 

angle  or  by  a  positive  angle.     Thus,  if  the  angle  XAP^  be  half 

TT 

a  right  angle,  the  direction  AP'  may  be  denoted  either  by  —  j 

or  +^- 

We  also  sometimes  admit  negative  as  well  as  positive  values 
of  the  radius  vector.     Thus,  suppose  the  co-ordinates  of  P  to 

TT  -r-r-  TT 

be  a  and  7 ;  that  is,  let  XAP=^,  and  AV=za  ;  if  we  produce 
PA  to  P'',  making  AP"  =  AP,  then  P''  may  be  determined  by 
saying  that  its  co-ordinates  are  —a  and  j.     The  radius  vector 


CO-ORDINATES   OF   A   POINT.  37 

is  considered  positive  when  it  is  measured  in  the  direction  of 
the  extremity  of  the  arc  measuring  the  variable  angle ;  it  is 
considered  negative  when  it  is  measured  in  the  oppo^te  direc- 
tion. 

Thus  the  co-ordinates 


r  and      j    represent  the  point  P, 
T  and      77  *^ 


4 


P5. 

p 

A 

Pi 

/^ 

-*-  G* 

t,  / 

^ 

"pT 

Pr 

/ 

P5 

^Pe 

P. 

— 7*  and 
—T  and      TT 
—7'  and      ^ 

-/'  and  -^  "  P. 

Thus  the  same  point  P  is  denoted  either  by  the  co-ordinates 

T  and  x>  or  —r  and  —^  or  —r  and  —  ^. 

Ex.  1.  Indicate  by  a  figure  the  position  of  the  point  whose 
co-ordinates  are  a^  15°,  where  a—\  inch. 
Ex.  2.  Indicate  by  a  figure  the  position  of  the  point  2^,  40°. 
Ex.  3.  Indicate  by  a  figure  the  position  of  the  points 

QTT  llTT  TT   TT 

— <2,45°;  —(^,—135°;  3<2,  ^;  ^a,-^;  2a  sin. -^j^. 

18.  Implicit  equations  of  a  point.  The  position  of  a  point 
may  be  determined  not  only  explicitly  by  co-ordinates,  but  im- 
plicitly by  means  of  simultaneous  equations  which  these  co- 
ordinates satisfy.  For  if  we  have  two  simultaneous  equations 
between  two  variables,  we  can  find  the  values  of  tliese  variables 
by  the  methods  of  Algebra,  and  these  values  are  the  co-Oi*di- 
nates  of  known  points. 

Ex.  1.  Thus,  suppose  we  have  the  equations 
2x-{-Sy=12,) 
Sx-2y=6,    ) 
we  find  x=3,  and  y=2. 


38  ANALYTICAL   GEOMETKY. 

Ill  this  and  the  following  examples  the  pupil  should  draw 
the  figure  representing  the  problem. 

Ex.  2.  Determine  the  point  whose  co-ordinates  satisfy  the 
equations  6x~-6y=d,    ) 

7x  —  6y=16.  ) 

Ans.  x=6,  and  ^=4. 

Ex.  3.  Determine  the  point  whose  co-ordinates  satisfy  the 
equations  x  ^  y 

a 


4-^-^ 
+  ^-^, 


0    a 


Zab 
Ans.  x=y= — -t- 
^     a+b 


Ex.  4.  Determine  the  points  whose  co-ordinates  satisfy  the 
equations  x-[-y=4:{x—y)^  \ 

x'-^lf  =  Z^.        S 

^715.(5,3),  and  (-5, -3). 
Ex.  6.  Determine  the  points  whose  co-ordinates  satisfy  the 
equations  ,  x^-\-xy=4zO,\ 

^  V    xy—'if=Q.    ) 

Ans,  (5,  3),  (-5,  -3),  (4^2,  ^/2\  and  (-4^/2,  -  V^). 

V 

19.  To  find  the  distance  of  any  ^oint  from  the  origin  in 
terms  of  the  co-ordinates  of  that  ^oint. 

Case  First.  Let  the  co-ordinates  be  rectan- 
gular. 


-X 


We  have  AF=AB^-f  BFr^cg^y^; 

therefore      AV=^x'~\-y''. 


-A.     Ji  Ex.  1.  Determine  the  distance  from  the  ori- 

gin to  the  point  whose  co-ordinates  are  x=Za^  y=^a. 

Ans.  AP=  V9^M^16^=5^. 

Ex.  2.  Determine  the  distance  of  the  point  --2^,  3 J,  from 
the  origin.  Ans.  5-v/13. 

Ex.  3.  Determine  the  distance  from  the  origin  to  the  point 
a  sin.  /3,  a  cos.  j3.  Ans.  a. 


CO-OEDINATES   OF   A  POINT.  39 

Ex.  4.  'Determine  the  distance  of  the  point  6a,  —3^,  from 
the  origin. 

20.  Case  Second.  "When  the  co-ordinates  are  oblique. 
•  From  P  draw  PD  perpendicular  to  AX;  -y/ 
then  (Geom.^  B.  lY.,  Prop.  13) 

AP2=AB2+BP2+2AB .  BD. 
But  by  Trig.,  Art.  41, 

K:cos.PBD::PB:BD.  ^       b 

Hence  BD=PB  cos.  PBD  (radius  being  unity). 

Therefore  AP^ = AB^ + BP^ + 2 AB .  PB  cos.  PBD. 
But  PBD^YAX,  which  we  will  represent  by  w. 

Hence         AP  =  (x^ + y^ + 2xy  cos.  w)^ . 

In  the  following  examples  we  will  suppose  the  axes  to  be 
inclined  at  an  angle  of  60°. 
•  Ex.  1.  Determine  the  distance  from  the  origin  to  the  point 

3«,4«.  ji^g^  AP={da^^lQa''-i-2W'xi)^=aVS7. 

Ex.  2.  Determine  the  distance  from  the  origin  to  the  point 
-2b,  Sh.  Ans.AP=hV7. 

Ex.  3.  Determine  the  distance  from  the  origin  to  the  point 
a  sin.  ft  a  cos.  (3.  ^^^^  ^(l_j_ ^  ^^^  2j3)*  ,; 

JVote.  Sin.  2A  =  2  sin.  A  cos.  A  (Trig.,  Art.  73).  ^ 

Ex.  4.  Determine  the  distance  from  the  origin  to  the  point 
6a,  —3a. 

21.  To  find  the  distance  between  ttoo  given  points. 

Case  First.  Let  the  axes  be  rectangular. 

Let  P  and  Q  be  the  two  points,  and  repre- 
sent the  co-ordinates  of  P  by  x^,  y^,  and  those 
of  Q  by  a?2,  y^,.  

Draw  PR  parallel  to  the  axis  of  x,  cutting     -^    ^        n 
m  in  E. 

Then  PQ2=PR2+PQ^ 

But  PR=MN=AK-AM=iC2-aJi 

and  QB3zQTN'-PM=2/2-yi. 


R 


4:0  ANALYTICAL    GEOMETEY. 

Therefore  PQ^  =  (x,-x,f  +  {i/,-y,)\ 

and  PQ  =.  V{x,-x,y^{y,-y,)\ 

Ex.  1.  Determine  the  distance  between  the  point  3,  4,  and 
the  point  4,  3.  Ans.  PQ2z=(3-4)H(4-3)2 .-.  PQ=  V2. 

Ex.  2.  Determine  the  distance  between  the  point  —3,4,  and 
the  point  4,  —3.  Ans.  1-\/2. 

Ex.  3.  Determine  the  distance  between  the  point  2,  2,  and 
the  point  —2,-2.  Ans.  4^2.  / 

Ex.  4.  Determine  the  distance  between  the  point  2a,  0,  and     / 
the  point  0,—2«.  Ans.  2a  V 2- 

Ex.  5.  Determine  the  distance  between  the  point  —  2a,  2a, 
and  the  point  4a,  —  Qa.        j, 

22.  Case  Second.  Let  the  axes  be  inclined  at 
^  an  angle  w.^v>v. 

Then,  as  in  AA  20, 
A.   -k    :k     ^        PQ^=PE2+RQ2+2PE.KQcos.YAX, 

or     PQ  =  V(«2-^i)^+(2/2-2/i)^+2(«2-^i)(y2-yi)  cos.  lo. 

Ex.  1.  Determine  the  distance  between  the  point  0,  3,  and 
the  point  4, 0, 

Ans.  PQ2=42+32_2.4.3  COS.  w  =  25-24  cos.  w, 
and  PQ  =  ^25- 24  cos.  to. 
Ex,  2.  Determine  the  distance  between  the  point  0,  3,  and 
the  point  —4,  0.  Ans.  -^25  + 24'  cos.  w. 

Ex.  3.  Determine  the  distance  between  the  point  2,  —2,  and 

the  point —2, 2.  4       r.    -     ^ 

A?is.  8  sm.  -^. 

Note.  2  sin.  2A=l-cos.^A  (Trig.,  Art.  74). 

Ex.  4.  Determine  the  distance  between  the  point  a,  0,  and 

the  point  0,  a.  ^       ^     .     w 

^  An^.  2a  sm.  -^. 

23.  Case  Third.  Let  the  co-ordinates  be  polar. 

p  Let  P  and  Q  be  the  two  given  points ;  repre- 

.Q    sent  AP  by  r^^  and  AQ  by  r^ ;  also  PAX  by  0^, 
^  and  QAX  by  02- 

^   -^      «  From  P  draw  PD  perpendicular  to  AQ. 


CO-ORDINATES   OF  A  POINT.  41 

By  Geom.,  Bk.  TV.,  Prop.  12, 

PQ2=AP2+AQ2-2AQxAD. 
But  AD= AP  cos.  PAQ  (radius  being  unity). 

Hence      PQ2= AP^^  AQ2-2AP  x  AQ  x  cos.  PAQ 

and  PQ  =  Vr^^-{-r^''-2r^r^  cos. (0^-0^). 

Ex.  1.  Determine  the  distance  between  the  point  2a,  30°,  and 
the  point  a,  60°.  Ans.  PQ^ = 4:0" +a^-  U^  x  W^, 

and  PQ  =aVo-2VS. 
Ex.  2.  Determine  the  distance  between  the  point  a,  0°,  and 
the  point  b,  30°.  A7i8.  TQ^=a^-{-b^--2abxiVS, 

and  PQ  =  Va'^+b'^-ab-x/S. 
Ex.  3.  Determine  the  distance  between  the  point  a,  0,  and 
the  point  —a,— 6. 

Ans.  PQ2 = a^ -\-a''-\-  2a''  cos.  26  =  2a\l  +  cos.  20), 
and  PQ  =  2a  cos.  6. 
Note.  2  COS.  2 A  =.  1  +  cos.  2 A  (Trig.,  Art.  74). 
Ex.  4.  Determine  tlije  distance  between  the  point  a,  0,  and 
the  point  a,  —  0.  v .,  A7is.  2a  sin.  0. 

24.  To  find  the  co-ordinates  of  the  jpoint  which  bisects  the 
straight  line  joining  two  given  joints. 

Let  D  be  the  point  required,  AN,  DN  its  co-ordinates,  and 
let  DK  cut  BF  in  E. 

Then 
AN=AL-fLN=AL+BE=AL+iBF; 

that  is,    AN=:^,+^^^=^i±^l 

In  Hke  manner,  im=^^^. 

Ex.  1.  Determine  the  co-ordinates  of  tlie  point  of  bisection 
of  the  line  joining  the  point  —1, 1,  w^ith  the  point  3,  —5. 

Ans.  x=l,  2/z=  — 2. 

Ex.  2.  Determine  the  co-ordinates  of  the  point  of  bisection 
of  the  line  joining  the  point  3,  —3,  with  the  point  5,-5. 


r^ 


42  ANALYTICAL   GEOMETKY. 

25.  To  find  the  area  of  a  triangle  whose  angular  joints 
are  given. 

Let  BCD  be  the  triangle,  and  let  the  co- 
ordinates of  B,  Cj  D  be  x^  y^^  Xr^  2/2?  ^3  2/3  ^'<^" 
spectivelj. 
The  area  BCD 
=rBCML+CD]SrM-BDNL. 
But         BCML=iLM(BL4-CM)z:rJ(^2-aj^)(2/2+2/i). 
So  also  CDNM=:J(^3-a'2)(y3+2/2), 

and  BDKL=-|(a?3— a?i)(2/3+2/i)- 

Therefore  the  area  BCD  = 
=i{fe-«^i)(y2+2/i)+fe-^2)(2/3+2/2)-(^3~^i)(2/3+yi)! 
= WiV^ + «^22/i + ^32/2  -  ^iVz  -  ^22/3  -  ^32/1)- 
Ex.  1.  Determine  the  area  of  the  triangle  whose  angular 
points  are  3, 4 ;  —  3,  —4 ;  0, 4.  Ans.  12. 

Ex.  2.  Determine  the  area  of  the  triangle  whose  angular 
points  are  0, 0 ;  1, 2 ;  2, 1.  a       ^ 

Ex.  3.  Determine  the  area  of  the  triangle  whose  angular 
points  are  a,0;  —a,0;  0, h.  Ans,  ah. 

Ex.  4.  Determine  the  area  of  the  triangle  whose  angular 

points  are  1, 1 ;  —  1,  2 ;  —1, 1. 

26.  To  convert  the  rectangular  co-ordinates  of  ajpoint  into 
jpolar  co-ordinates^  and  vice  versa. 

Let  X  and  y  denote  the  co-ordinates  of  P  referred  to  the  rect- 
Y  _^  angular  axes  AX  and  AY.     Also,  let  r  and  0 

denote  the  polar  co-ordinates  of  P,  the  pole 
being  at  the  origin  A,  and  AX  being  the  initial 
line.     Draw  PD  perpendicular  to  AX.    Then, 
by  Trig.,  Art.  41, 

AD=:AP  COS.  PAD,    or  x=r  cos.  0; 
also  PD=rAP  sin.  PAD,     or  y=:r  sin.  0, 

which  equations  enable  us  to  deduce  the  rectangular  co-ordi- 
nates of  a  point  from  the  polar  co-ordinates.  ^ 

I): 


CO-ORDINATES   OF  A  POINT-  43 

Again,  AD^  +  PD^ = AP^,  or  x^-\-y'= r^, 

and  AD :  K : :  PD :  tang.  PAD,    or  ^=tang.  0, 

which  equations  enable  us  to  deduce  the  polar  co-ordinates  of 
a  point  from  the  rectangular  co-ordinates. 

Ex.  1.  Find  the  polar  co-ordinates  of  the  point  whose  rect- 
angular co-ordinates  are  x=l,y=l,  and  indicate  the  point  by 
a  figure.  Ans.  r—  -\/2,  0=45°. 

Ex.  2.  Find  the  polar  co-ordinates  of  the  points  whose  rect- 
angular co-ordinates  are 

(1)  x^-\,  2/=-!- 2. 

(2)  x^-\,  y=-± 

(3)  ^=-1-1,  y=-± 

Ex.  3.  Find  the  rectangular  co-ordinates  of  the  point  whose 

polar  co-ordinates  are  r= 3,  0=o-  ,  3        3,^ 

3  Ans.x=^,y=^-y/Z. 

Ex.  4.  Find  the  rectangular  co-ordinates  of  thq  points  whose 
polar  co-ordinates  are 


(l)r= 

:+3, 

6  = 

~3- 

(2)^= 

:-3, 

e= 

+1- 

(3)^:. 

:-3, 

0= 

""3* 

u 


ANALYTICAL   GEOMETKY. 


sectio:n  il 

THE   STRAIGHT  LINE. 

23.  Definition.  The  equation  of  a  line  is  the  equation 
which  expresses  the  relation  l)etween  the  two  co-ordinates  of 
every  point  of  that  line. 

Hence,  if  any  point  be  taken  upon  the  line,  and  the  values 
of  X  and  y  for  that  point  be  substituted  in  the  equation,  the 
equation  will  be  satisfied ;  and  conversely,  if  the  co-ordinates 
of  any  point  whatever  of  the  plane  satisfy  the  equation  of  a 
line,  that  point  will  be  on  the  line. 

29.  To  find  the  equation  to  a  straight  line  referred  to  rect- 
angular axes. 

Let  A  be  the  origin  of  co-ordinates, 

AX  and  AY  be  rectangular  axes,  and 

let  PC  be    any  straight  line  whose 

equation  is  required  to  be  determined. 

Take  any  point  P  in  the  given  line, 

and  draw  PB  parallel  to  AY;  then 

will  PB  be  the  ordinate  and  AB  the  abscissa  of  the  point  P. 

From  A  draw  AD  parallel  to  CP,  meeting  the  line  BP  in  D. 

Let  KB=x, 

BP=y, 
tang.PEXorDAX=m,  ^^ 


and  ACorDP=:c. 

Then,  by  Trigonometry,  Theorem  IL,  Art.  42, 
ABtBBm  radius 
that  is,  X :  BD : :  1 

or  BD=m£c. 

But  BP=BD-fDP; 

that  is,  y — mx  -f  c. 


tang.  D  AX  • 


l> 


THE    STRAIGHT   LINE. 


45 


Hence  the  equation  to  a  straight  line  referred  to  rectangular 
axes  is  y—mx-^-c; 

where  x  and  y  are  the  co-ordinates  of  any  point  of  the  line,  m 
represents  the  tangent  of  the  angle  which  the  line  makes  witli 
the  axis  of  abscissas,  and  c  the  distance  from  the  origin  at 
which  it  intellects  the  axis  of  ordinates. 


^y 


30.  Signs  qfm  and  c. 

If  the  line  CP  cuts  the  axis  of  ordi- 
nates below  the  origin,  then  c  or  AC  will 
be  negative. 

In  that  case,  BP=BD-DP ; 
or,  y=7nx—c. 

The  angle  which  the  line  makes  with 
the  axis  of  abscissas  is  supposed  to  be  measured  from  the  axis 
AX  around  the  circle  by  the  left.  If  the 
line  CP  is  directed  downward  toward  the 
right,  as  in  the  annexed  figure,  the  line 
makes  either  an  obtuse  angle,  CEX,  with 
the  axis  of  abscissas,  or  the  negative  acute 
angle  CEA,  the  tangent  of  either  of  which 
angles  is  negative  (Trig.,  Art.  69). 

In  this  case  we  have 

AB :  BD : :  radius :  tang.  DAX, 
or  X :  BD : :     1      :  m. 

The  tangent  of  DAX  being  negative,  BD  is  also  negative. 

But  BP=-BD+DP, 

and  the  equation  becomes  y=  —mx+c, 

where  it  must  be  observed  that  the  minus  sign  applies  only  to 
the  quantity  m,  and  not  to  x,  for  the  sign  of  x  depends  upon  its 
direction  from  the  origin  A. 

If  the  line  CP  is  directed  downward  toward 

the  right,  and  cuts  the  axis  of  ordinates  below 

the  origin,  then  c  is  negative  as  well  as  m  /  and 

since  BP=— BD— DP,  the  equation  becomes 

y=—mX'-c. 


46 


ANALYTICAL   GEOMETRY. 


It  is  to  be  remembered  that  the  symbols  a?,  y,  m,  and  c  may 
stand  for  negative  numbers,  and  therefore  the  single  equation 

y=mx-{-G 
may  represent  any  line  whatever. 

31.  Four  diferent  positions  of  a  line.  There  may,  there- 
fore, be  four  positions  of  the  proposed  line,  and  these  positions 
are  indicated  by  the  signs  of  m  and  g  in  the  general  equation. 

1.  Let  the  line  cut  the  axis  of  X  to  the 
left  of  the  origin,  and  the  axis  of  Y  above 
it ;  then  m  and  c  are  both  positive,  and 
the  equation  is 

y^-\-mx-\-c. 

2.  If  the  line  cuts  the  axis  of  X  to  the 
right  of  the  origin,  and  the  axis  of  Y  be- 
low it,  then  m  will  still  be  positive,  but 
c  will  be  negative,  and  the  equation  be- 
comes y=-\-  ^^ — c. 

3.  If  the  line  cuts  the  axis  of  X  to  the 
right  of  the  origin,  and  the  axis  of  Y 
above  it,  then  m  becomes  negative  and  c 
positive.  In  this  case,  therefore,  the  equa- 
tion is  y——mx-\-c. 

4.  If  the  line  cuts  the  axis  of  X  to  tlie 
left  of  the  origin,  and  the  axis  of  Y  below 
it,  then  both  m  and  c  will  be  negative,  so 
-x   that  the  equation  becomes 
2/=— ma?— c. 
If  we  suppose  the  straight  line  to  pass 
through  the  origin  A,  then  c  will  become 
zero,  and  the  general  equation  becomes 

y=imx, 
which  is  the  equation  of  a  straight  line  passing  through  the 
origin. 


p 

X 


THE    STEAIGHT   LINE.  47 

32.  Direction  of  a  line  indicated.  It  will  be  seen  that  the 
direction  of  the  proposed  line  is  indicated  by  the  symbol  m. 
If  m  is  very  small  and  positive,  the  line  whose  p"  y 
equation  is  y  —  mx  takes  the  position  AP, 
near  the  axis  AX.  As  m  increases  the  line 
changes  its  position,  and  when  m=l  the  line 
makes  an  angle  of  45°  with  AX.  As  the 
value  of  m  increases  the  line  approaches  AY,  and  coincides 
with  it  when  m  becomes  infinite. 

If  m  is  negative  and  very  large,  the  line  assumes  the  position 
AP^^,  and  as  m  decreases  the  line  moves  toward  AX',  and 
when  m=  —  1  the  line  bisects  the  angle  YAX'.  When  m  be- 
comes zero,  the  line  coincides  again  with  the  axis  of  abscissas. 

So,  also,  if  the  point  P  is  supposed  to  travel  round  A  through 
the  third  and  fourth  quadrants,  the  value  of  m  will  be  positive 
in  the  third  quadrant  and  negative  in  the  fourth. 

Ex.  1.  Let  it  be  required  to  draw  the  line  whose  equation  is 
2/  =  2«+4. 

Draw  the  co-ordinate  axes  AX,  AY. 
Now  if  in  this  equation  we  suppose  a?==0, 
the  value  of  y  will  designate  the  point  in 
which  i\\Q  line  intersects  the  axis  of  ordi- 
nates,  for  this  is  the  only  point  of  the  line 
whose  abscissa  is  zero.  This  supposition 
will  give  y—^' 

Hence,  if  we  take  AB=4,  B  will  be  one  point  of  the  required 
line. 

Again,  if  in  the  proposed  eqnation  we  suppose  2/=0,  the 
value  of  X  which  is  found  from  the  equation  will  designate 
the  point  in  which  the  line  intersects  the  axis  of  abscissas,  for 
that  is  the  only  point  of  the  line  whose  ordinate  is  zero.  This 
supposition  will  give 

2aj=-4, 
or  a?=:— 2. 

Hence,  if  we  lay  off  from  A  toward  the  left  a  distance  AC 
=  2,  C  will  be  a  second  point  of  the  proposed  line.     Draw  the 


48 


ANALYTICAL   GEOMETKY. 


straight  line  BC,  and  produce  it  indefinitely  both  ways ;  it  will 
be  the  line  whose  equation  is  y=z2x+4:. 

The  student  should  regard  every  algebraic  equation  in  this 
treatise  as  expressing  some  geometrical  truth,  and  he  should 
accustom  himself  to  express  these  truths  in  appropriate  geo- 
metrical language.  Thus  the  equation  y=2x+4:  expresses  the 
truth  that  a7iy  ordinate  of  a  certain  straight  line  is  equal  to 
twice  the  corresponding  abscissa^  increased  hy  four. 

So  also  the  general  equation  of  a  straight  line,  y=mx-\-c, 
expresses  the  truth  that  any  ordinate  of  any  straight  line  is 
equal  to  some  multiple  of  the  corresponding  abscissa,  in- 
creased hy  a  constant  number , 


33.  Any  number  of  points  of  a  line  determined.  When  the 
equation  of  a  line  is  given,  we  may,  if  desired,  determine  any 
number  of  points  of  the  line  by  assuming  particular  values  for 
a?,  and  computing  the  corresponding  values  of  y. 

Thus,  if  in  the  equation  2/= 2a?  4-4  we  suppose 


1,  we  find  y—  6. 


X=z% 

x^S, 

X  =  4r, 


a?=— 1,  we  find  ?/—     2. 

yz^  8.  x=-2,         "     y^     0. 

y=10,  x=^3,        "     2/== -2. 

?/=12,  etc.  aj=— 4,  "  ?/=— 4,  etc. 
In  order  to  represent  all  these  values  by  a 
figure,  set  off  on  the  axis  of  abscissas  lines 
equal  to  1, 2, 3,  etc.,  botli  to  the  right  and  left 
of  A ;  then  erect  a  perpendicular  from  each 
of  these  points,  and  make  it  equal  to  the  cor- 
responding value  of  y,  setting  it  off  above 
AX  if  the  ordinate  be  positive,  but  below 
AX  if  negative.  Tlie  required  line  must 
pass  through  all  the  points  thus  determined. 


34.  Variables  and  constants.  In  the  equation  y=mx-\-c, 
m  and  c  remain  unchanged  so  long  as. we  consider  the  same 
straight  line  ;  they  are  therefore  called  constant  quantities,  or 
constants.     But  x  and  y  may  have  an  indefinite  number  of 


THE    STRAIGHT   LINE.  49 

values,  since  we  may  ascribe  to  one  of  them,  as  aj,  any  value 
we  please,  and  find  from  the  equation  the  corresponding  value 
of  y.  X  and  y  are  therefore  called  variable  quantities,  or  vari- 
ahles. 

35.  Meaning  of  the  equation  of  a  line.  The  equation  of  a 
line  may  be  regarded  as  a  statement  of  some  geometrical  prop- 
osition respecting  that  line. 

Thus  the  equation 

may  be  regarded  as  the  algebraic  statement  of  the  proposition, 
any  ordinate  of  a  certain  line  is  always  equal  to  twice  its  cor- 
resjponding  abscissa  increased  by  ten. 

36.  Equation  to  a  line  jparallel  to  one  of  the  axes.  If  in  the 
equation  y=mx-\-c  we  suppose  m=0,  the  line  will  be  parallel 
to  the  axis  of  X,  and  the  equation  becomes 

y=0.x-\-c, 
or  y=c. 

This  is  then  the  equation  of  a  line  parallel  to  the  axis  of  X. 
If  c  is  positive,  the  line  is  above  the  axis  of  X ;  if  negative,  it 
is  below  it. 

So  also  x=±:a  is  the  equation  to  a  straight  line  parallel  to 
the  axis  of  Y. 

Examjples.  Construct  the  lines  of  w^hich  the  following  are 
the  equations : 

1. 2/=2iK+3.  4.  3^=— 2aj— 5.  7. 2/= 6. 

2.  y=Zx-n.  5.  y=3a?.  8.  y=.  -2. 

3.  j/=— £C-|-2.  6. 2/=aj.  9. 2/=— a?. 

37.  Every  equation  of  the  first  degree  containing  two  vari- 
ables represents  a  straight  line.  '  '      < 

Every  equation  of  the  first  degree  containing  two  variables 
can  be  reduced  to  the  form 

Aaj+By+C^O, 
in  which  A,  B,  and  C  may  be  positive  or  negative.    We  shall 

C 


50  ANALYTICAL   GEOMETEY. 

now  prove  that  every  equation  of  this  form  represents  a  straight 
line. 

Y  In  this  equation  put  2/=0,  and  we  have 

C 
-X  ^=~T?  which   represents   the   point  D 

where  the  line  intersects  the  axis  of  X. 

Again,  pnt  x=0,  and  we  have  ?/=— v^, 

which  represents  the  point  E  where  the  line  intersects  the  axis 
of  Y.  We  have  thus  determined  two  points  in  the  line  which 
this  equation  represents. 

Let  P  be  any  other  point  of  the  line  or  curve  represented  by 
the  given  equation.  "We  are  to  prove  that  P  is  on  the  straight 
line  passing  through  the  points  D  and  E. 

Since  P  is  supposed  to  be  on  the  line  represented  by  the 

given  equation,  its  co-ordinates  must  satisfy  this  equation ;  and 

representing  its  co-ordinates  by  x  and  y,  we  shall  have 

Ax+B2j-\-C=0, 

-Q-Ax    ^^ 
whence  y-= ^ =PR  g-, 

C       C         C         -C-Aa?  .^     ^.    ,, 
JN  ow   ~"  T  •  —  t5  *  *  —  x  —  ^  • T> J  identically. 

But  these  several  terms  are  equal  to  those  of  the  proportion 

AD:AE::DK:PK; 

that  is,  PK  is  a  fourth  proportional  to  the  three  lines  AD,  AE, 

and  DK;  that  is,  P  lies  on  the  straight  line  joining  D  and  E, 

and  the  equation  AaJ+By+C=0  represents  that  straight  line. 

If  either  A,  B,  or  C  be  negative,  the  same  demonstration  will 
apply  with  a  slight  change  of  the  figure.   ' 

This  equation  always  represents  so7ne  straight  line,  and  may 
be  made  to  represent  any  one  by  giving  appropriate  values  to 
A,  B,  and  C. 

If  in  this  equation  A=0,  then  the  line  is  parallel  to  the  axis 
of  X ;  if  B=:0,  the  line  is  parallel  to  the  axis  of  y ;  if  C=0, 
the  line  passes  through  the  origin. 

Examjples,  Draw  the  straight  lines  represented  by  the  fol- 
lowing equations : 


D 


THE    STEAIGHT   LINE.  51 

1.  a;+2/+10=0.         6.2y=3x-6.  11.  x=2?/. 

2.  x+y=10,  7.  y=4:^x.  12.  x=4:. 

3.  x-{-y=0.  8. 2«=?/4-7.  13. 2/=2. 

X     ?y 

4.  2aj  +  3?/r:=0.  9.2+^=1.  14. 4«— 3y=l. 

5.  4aj4-3y=l.  10. 2/~3  =  2(aj-2).      15.  x-2y=--4:. 

38.  T(9  j^/i(^  ^^6  equation  to  a  straight  line  which  jpasses 
through  a  given  point 

When  a  point  P  is  not  completely  determined,  its  co-ordi- 
nates are  denoted  by  the  variables  x  and  y ;  but  when  the  po- 
sition of  a  point  is  completely  known,  the  co-ordinates  are  gen- 
erally denoted  by  the  letters  a,  b,  or  by  x,  y,  with  suffixes,  as  x^, 
Vv  ^2?  2/2  ?  or  by  cc  and  y  with  accents,  as  x\  y\  x'\  y'\  etc. 

Let  PCE  be  the  straight  line,  C  the  given         y 
point  whose  co-ordinates  are  x^^y^^  and  P 
any  point  of  the  line  whose  co-ordinates  are 
X  and  y.    Draw  the  ordinates  CL,  PM ;  also 
draw  CD  parallel  to  AX. 

Now  PD=?/  — ?/j,  /  ijl ^   ^-y^sp'  -^    ^ 

and  CD=x~Xy  7-^**  ^(J—iO 

But  CD  :PD::  radius:  tang.  PCD. 

PD 
Hence  pyr=tang.  PCD,  which  we  will  represent  by  m, 

which  is  the  equation  of  a  straight  line  passing  through  a  given 
point  P. 

Since  the  coefficient  m,  which  fixes  the  direction  of  the  line, 
is  not  determined,  there  may  be  an  infinite  number  of  straight 
lines  drawn  through  a  given  point.  This  is  also  apparent  from 
the  figure. 

39.  Zinc  passing  through  a  given  point  and  parallel  to  a 
given  line.  If  it  be  required  that  the  line  shall  pass  through 
a  given  point,  and  make  a  given  angle  with  the  axis  of  X,  then 


M 


52  ANALYTICAL  GEOMETRY. 

m  becomes  a  known  quantity,  and  if  we  put  nn'  for  tlie  tangent 
of  the  given  angle  we  shall  have 

w^hich  is  the  equation  of  a  straight  line  passing  through  a  giv- 
en point,  and  making  a  given  angle  with  the  axis  of  X. 

Ex.  1.  Draw  a  line  through  the  point  whose  abscissa  is  5  and 
ordinate  3,  making  an  angle  with  the  axis  of  abscissas  whose 
tangent  is  equal  to  2,  and  give  the  equation  of  the  line. 

Ans.  The  equation  is  y—2a?4-Y— 0. 

Ex.  2.  Find  the  equation  to  the  straight  line  which  passes 
through  the  point  («,  J),  and  makes  an  angle  of  30°  with  the 
axis  of  X.  Ans.x—a—{y—'l))^J'^. 

Ex.  3.  Find  the  equation  to  the  line  which  passes  through 
the  point  (4, 4),  and  makes  an  angle  of  45°  with  the  axis  of  X. 

(  J  40.  To  find  the  equation  to  the  straight  line  which  passes 
through  two  given  points. 

Let  B  and  C  be  the  two  given  points, 
the  co-ordinates  of  B  being  x^  and  y^^  and 
the  co-ordinates  of  0  being  x^  and  y^. 
Then,  since  the  general  equation  for  ev- 
ery point  in  the  required  line  is 

y-mx\c,  (1) 

it  follows  that  when  the  variable  abscissa  x  becomes  a?j,  then  y 
will  become  y^ ;  hence 

y^^mx^-^c.  (2) 

Also,  when  the  variable  abscissa  x  becomes  ajg?  ^^^^  V  ^^' 
comes  2/2)  and  hence         y^-=mXr>^-\-c.  (3) 

By  combining  these  three  equations  we  may  eliminate  m 
and  c. 

If  we  subtract  equation  (2)  from  equation  (1),  we  obtain 

y^y^  =  m{x-x;).  \  (4) 

Also,  if  we  subtract  equation  (2)  from  equation  (3),  we  ob- 
tain 2/2-2/i  =  ^fe-»^i)> 


from  which  we  find  m= 


y2-y 


THE   STKAIGHT   LINE. 


53 


-    Substituting  this  value  of  m  in  equation  (4),  we  have   ^^oj^-^ 

which  is  the  equation  of  the  line  pass^  thi'ough  the  two  given 
points  B  and  C. 

It  is  evident  from  the  figure  that  — — —  denotes  the  tangent 
of  the  angle  BCD  or  BEX.  ^^"^^ 

If  the  origin  be  one  of  the  proposed  points  (a?2,  y^,  then 
aj2=0  and  ^2=^5  ^^^  the  equation  becomes 

2/1 


y- 


X, 


which  is  the  equation  to  a  straight  hne  passing  through  the  or- 
igin and  through  a  given  point. 

Ex.  1.  Find  the  equation  to  the  straight  line  which  passes 
through  the  two  points  whose  co-ordinates  are  a?^  =  7, 2/^  =  4, 
and  x^—^,  1/2  —  ^9  ^^d  determine  the  angle  which  it  makes 
with  the  axis  of  abscissas. 

Ex.  2.  Find  the  equation  to  the  straight  line  which  passes 
through  the  two  points  x^  =  2,  y^  =  S,  and  x^=4, 2/2=^- 

Ex.  3.  Find  the  equations  to  the  straight  lines  which  pass 
through  the  following  pairs  of  points  : 


5. 


(1) 

^i=3j2/i=4; 

and 

x^=l,y^=2. 

(2) 

x,  =  5,7/^  =  6; 

a 

%^-l,  2/2=0. 

(3) 

x,  =  l,y^  =  2; 

a 

^2  =  2, 2/2= -4. 

(4) 

^i=4,2/i  =  -2; 

u 

^2^-3,^2=- 

(5) 

«^i  =  3,2/,=  -2; 

u 

^2  =  ^>^2  =  ^- 

(6) 

«'i  =  2,y^  =  5; 

ii 

o^2=^,y2=-'^' 

(^) 

x,  =  0,y,  =  l; 

a 

«^2  =  l,y2=-l- 

(8) 

^i=0,y,=  -a; 

a 

^2  =  0,2/2=-^. 

(9) 

x^  =  a,7/^  =  b; 

a 

^2=A2/2=-^- 

10) 

x,  =  a,y,  =  -b; 

a 

^2=-(^,y2=- 

41.  Definition.  The  distance\  from  the  origin  to  the  point 
where  a  line  intersects  the  axis  of  X  is  called  the  intercept  on 
the  axis  of  X;  and  the  distance  from  the  origin  to  the  point 


54 


ANALYTICAL    GEOMETRY. 


where  a  line  intersects  the  axis  of  Y  is  call- 
ed the  intercept  on  the  axis  of  Y. 

Thus,  in  the  annexed  figure,  AB  and  AC 
are  the  intercepts  of  the  line  PC  on  the  two 
axes. 


42.  To  find  the  equation  to  a  straight  line  in  terms  of  its 
intercejpts  on  the  two  axes. 

Let  B  and  C  be  the  points  where  the 
straight  line  meets  the  axes  of  y  and  x 
respectively.  Suppose  AC==^,  and  AB 
=  h.  Let  P  be  any  point  in  the  line,  and 
let  X  and  y  be  its  co-ordinates.  Draw 
PD  parallel  to  AY.     Then,  by  similar 


c^ 


rX 


triangles,  we  have 

that  is, 
whence 


AB:DP::AC:DC 

h:y::a:a—x, 
X    y    ^ 


which  is  the  equation  to  a  straight  line  in  terms  of  its  inter- 
cepts a  and  h. 

Ex.  1.  Find  the  equation  to  a  straight  line  which  cuts  off  in- 
tercepts on  the  axes  of  x  and  y  equal  to  3  and.  —5  respectively. 

Ex.  2.  Find  the  equation  to  a  straight  line  w^hich  cuts  off  the 
intercepts  —4  and  2. 

43.  To  find  the  angle  included  between  two  given  straight 
lines. 

Let  BC  and  DE  be  any  two  lines  in- 
tersecting each  other  in  P.     Let  the 
equation  to  the  line  BC  be 
y=m^x-\-c^, 
and  the  equation  to  the  line  DE  be 
y=zmrp^-c^\ 

then  m^  will  be  the  tangent  of  the  angle  BCX,  and  mg  the  tan- 
gent of  the  angle  DEX.    Now,  because  PCX  is  the  exterior 


THE   STEAIGHT  LINE.  65 

angle  of  the  triangle  PEC,  it  is  equal  to  the  sum  of  the  angles 
CPE  and  PEC ;  that  is,  the  angle  EPC  is  equal  to  the  differ- 
ence of  the  angles  PCX  and  PEX,  or 

EPC=PCX-PEX, 
whence  tang.  EPC  =  tang.  (PCX-PEX), 

which,  by  Trig.,  Art.  77, 

tang.  PCX-tang.  PEX 
- 1  +  tang.  PCX  X  tang.  PEX 

which  denotes  the  tangent  of  the  angle  included  between  the 
two  given  lines. 

44.  To  determine  the  co-ordinates  oftJiejpoint  of  intersec- 
tion of  two  given  straight  lines. 

Let  tlie  equation  to  one  line  be 

2/=m,aj+^i,  (1) 

and  the  equation  to  the  other 

y^m^x+Cr,.  (2)     , 

Since  the  co-ordinates  of  ever3''  point  on  a  line  must  satisfy 
its  equation,  the  co-ordinates  of  the  point  through  which  both 
the  lines  pass  will  satisfy  both  equations ;  we  must,  therefore, 
find  the  values  of  x  and  y  from  (1)  and  (2)  regarded  as  simul- 
taneous equations.     We  thus  obtain 

c,—c^         -,        c/in^—CMi.. 
x-—^ 2_  andv=:-      ^     ■- 


7ri^—7}i'        "        TYi^—m 


which  are  the  co-ordinates  of  the  point  of  mtersection  of  the 
two  lines. 

Ex.1.  Find  the  angle  included  between  the  lines  x-\-y—l 
and  2/=:a?+2;  also  find  the  co-ordinates  of  the  point  of  inter- 

^/i5.  90°,  a?=-^'2/=2* 
Ex.  2.  Find  the  angle  between  the  lines  x  +  Sy=:l  and  x—2y 
=  1 ;  also  the  co-ordinates  of  the  point  of  intersection. 

Ans.  45°,  x=l,  y=0. 
Ex.3.  Find  the  angle  between  the  lines  x-{-y\/S  =  0  and 


56  ANALYTICAL   GEOMETRY. 

x—y\/S=2;  also  the  co-ordinates  of  the  point  of  intersec- 
*'^°-  Ans.  60°,  x=l,y=-f. 

Ex.  4.  Find  the  angle  between  the  lines  Sy— a?=0  and  2x+y 
=  1 ;  also  the  co-ordinates  of  the  point  of  intersection. 

Ans.  81°  62',  x=^,  y^\, 

Ex.  5.  Find  the  angle  between  the  lines  3?/— 2a?4-l  =  0  and 
3aj— ^=0;  also  the  co-ordinates  of  the  point  of  intersection. 

Ex.  6.  Find  the  angle  between  the  lines  a?+y— 3  =  0  and 
a?+2/=2 ;  also  the  co-ordinates  of  the  point  of  intersection. 

45.  To  find  the  equation  to  the  straight  line  which  passes 
through  a  given  point,  and  is  perpendicular  to  a  given 
straight  line. 

Let  x^y^  be  the  co-ordinates  of  the  given  point,  and 
y=mx-\-G 
the  equation  to  the  given  line.     The  form  of  the  equation  to  a 
line  through  {x^y^  (Art.  38)  is 

y-y^=m^(x-x^). 
The  tangent  of  the  angle  between  the  two  lines  is  (Art.  43) 


1  +  mm^* 
If  the  angle  of  intersection  of  the  two  lines  be  a  right  angle, 
its  tangent  must  be  infinite,  and  the  denominator  l+?7i7?2j  must 
become  zero,  so  that  we  must  have 

1 

m,  =  — . 
^         m 

Hence  the  required  equation  is 

which  is  the  equation  to  the  straight  line  passing  through  the 
point  (a?i2/J,  and  perpendicular  to  the  line  y~mx-\-c. 

46.  Condition  of  perpendicularity.    We  conclude  from  the 
last  article  that  y=^—-\-c. 


THE   STRAIGHT   LINE. 


67 


represents  a  line  perpendicular  to  tlie  line 

y=zmx-{-c. 
The  condition  by  which  two  straight  lines  are  shown  to  be 
at  right  angles  to  each  other  may  also  be  determined  as  follows : 

Let  BC  be  a  given  line,  and  let  tang. 
BCX=m. 

Let  DE  be  perpendicular  to  BC,  and 
let  tang.  DEX^mj ;  then 

tang.  DEX=-tang.  DEA, 

=  —  cotang.  BCA ; 
1 


that  is, 


m,  = .     (Trig.,  Art.  28.) 

1  qn       ^        ^^  ^ 


Hence  we  see  that  when  two  lines  are  perpendicular  to  each 
other,  the  tangents  of  the  angles  which  they  make  with  either 
axis  are  the  reciprocals  of  each  other ,  and  have  contrary  signs, 
-  Ex.  1.  Find  the  equation  to  the  line  which  passes  through 
the  origin,  and  is  perpendicular  to  the  line  x-\-y=2. 

Ans.  y—x. 

Ex.  2.  Find  the  equation  to  the  line  which  passes  through 
the  point  a?j  =  2, ?/^=— 4,  and  is  peq^endicular  to  the  line  3y 
+  2£C— 1=0.  Ans.'iy^Zx-X^. 

Ex.  3.  Find  the  equation  to  the  line  which  passes  through 
the  point  (8, 4),  and  is  perpendicular  to  the  line  whose  equation 
is  2/= 2a?— 16. 

Ex.  4.  Find  the  equation  to  the  line  which  passes  through 
the  point  (—1,  3),  and  is  perpendicular  to  the  line  3a; +4?/ +2 
=  0. 

-^      47.  To  find  the  perpendicular  distance  of  a  given  point 
from  a  given  straight  line. 

Let  P  be  the  given  point,  whose  co-ordi- 
nates are  x^^,  and  let  BC  be  the  given  straight 
line  whose  equation  is 

y=LmjX-\-c. 
From  P  draw  PD  perpendicular  to  BC, 
and  PM  perpendicular  to  AX,  cutting  BC  in 

C2 


68  ANALYTICAL    GEOMETRY. 

E.    ]^ow,  since  the  above  equation  applies  to  every  point  of 
BC,  it  must  apply  to  E ;  that  is, 

The  perpendicular  PD  =  PE  sin.  PED. 

But  FE=FM.-.ME=y,-mx,-^c,   Piixv^,'^^->V,~.c 

and  sin.  PED^sin.  CEM=cos.  ECM= ^vw= 

sec.  ECM 

1  1 


Vl  +  (tang.  ECMy     Vl + m' 


Therefore  PD    '^~''''' 


Vl  +  m'  ' 

which  equation  expresses  the  distance  from  the  given  point 
(^i^i)  t^  ^^^  given  straight  line. 
If  the  point  P  be  at  the  origin,  then  x^  =  0,  y-^  —  O,  and  we 

have  PD:=     ~^    :, 

which  equation  expresses  the  distance  of  the  proposed  line 
from  the  origin. 

Ex.  1.  Find  the  perpendicular  distance  of  the  point  2, 3  from 
the  line  x-\-y=l.  Ans.  2-^2. 

Ex.  2.  Find  the  distance  of  the  point  —1,  3  from  the  line 

5 

Ex.  3.  Find  the  distance  of  the  point  0, 1  from  the  line  a?— 3y 

=1.  ,       2^/10 

Ans.  — ? — . 
5 

Ex.  4.  Find  the  distance  of  the  point  3,  0  from  the  line 
-+1=1.  Ans. 


2^3-^-  "Vis* 

Ex.  5.  Find  the  distance  of  the  point  1,-2  from  the  line 

aj-fy— 3  =  0.  A71S.  2V2. 

Ex.  6.  Find  the  distance  of  the  origin  of  co-ordinates  from 

the  Ime  ^+^=1.  Ans,  -tto. 


THE   STRAIGHT   LINE.  69^ 

Ex,  7.  Find  the  distance  of  the  point  3,-5  from  the  line 
2aj-.  8^+7=0. 

Ex.  8.  Find  the  distance  of  the  point  8, 4  from  the  line  y^'^x 
-16. 

48.  To  find  the  equation  to  a  straight  line  referred  to  ob- 
lique axes. 

Let  A  be  the  origin  of  co-ordinates ;  let 
AX,  AY  be  the  oblique  axes,  and  let  PC 
be  any  straight  line  whose  equation  is  re- 
quired to  be  determined.  Take  any  point 
P  in  the  given  line,  and  draw  PB  parallel 
to  AY ;  then  will  PB  be  the  ordinate  and  :e  a  5" 
AB  the  abscissa  of  the  point  P.  Through  the  origin  draw  a 
line  AD  parallel  to  CP,  meeting  the  line  BP  in  D. 

Denote  the  inclination  of  the  axes  by  w,  and  the  angle  DAX 
by  a.  Since  PB  is  parallel  to  AY,  the  angle  ADB  is  equal  to 
DAY;  that  is,  equal  to  a>— a. 

Let  a?,  y  be  the  co-ordinates  of  P,  and  represent  AC  or  DP 
by  G. 

Then,  by  Trig.,  Art  49, 

BD :  AB : :  sin.  a :  sin.  (o>— a). 

Hence  BD=-; — j-^ — r. 

sm.  (w  — a) 

But  BP=BD-fDP. 

X  sin.  a 

Hence  y = — — -r r  -\-  c. 

^    sm.  (w— a)      ' 

which  is  the  equation  to  a  straight  line  referred  to  oblique 

axes.  ^  ''■  ■■-'    vO  -    ''  ■ 

H  we  put  m  for  -: — -f- r,  the  equation  becomes 

^  sm.  (w— a)'  ^ 

y=mx-\-c, 
which  is  of  the  same  form  as  the  equation  referred  to  rectan- 
gular axes.  Art.  29.     The  meaning  of  c  is  the  same  as  in  Art. 
29 ;  but  the  factor  m  denotes  the  ratio  of  the  sine  of  the  incli- 
nation of  the  line  to  the  axis  of  X,  to  the  sine  of  its  inclination 


60  ANALYTICAL   GEOMETRY. 

to  the  axis  of  Y.    When  the  axes  are  at  right  angles  to  each 
other,  m  becomes  the  tangent  of  a. 

49.  To  find  the  polar  equation  to  a  straight  line. 
Let  BC  be  any  straight  line,  and  P  any 

point  in  it.  Let  A  be  the  pole,  AX  the  in- 
itial line,  and  let  AD  be  drawn  from  A 
perpendicular  to  BC.  Let  AD  =j!?,  the  an- 
gle DAX=:a,  and  let  the  polar  co-ordinates 
of  P  be  r,  0 ;  then  we  shall  have 
AD = AP  COS.  PAD ;  Qap  >.  ^^-  <l^  0^ 
that  is,  ^=:^  COS.  (0— a), 

or  7"=^  sec  (0— a), 

which  is  the  polar  equation  to  a  straight  line. 

If  AD  be  taken  for  the  initial  line,  then  a=0,  and  the  equa- 
tion becomes  r=p  sec.  ^, 

which  is  the  equation  to  a  right  \mQ  jperpendicular  to  the  ini- 
tial line. 

To  trace  a  right  line  by  its  polar  equation,  we  find  its  inter- 
cept on  the  initial  line  by  making  0=0.  Then  from  the  pole 
as  a  centre,  with  a  radius  equal  to^,  describe  a  circle,  and 
draw  a  tangent  to  this  circle  from  the  point  first  determined ; 
this  tangent  line  will  be  the  line  required. 

50.  To  find  the  polar  equation  to  a  line  passing  through 
the  pole. 

Let  X  and  y  denote  the  co-ordinates  of  P  re- 
ferred to  rectangular  axes  ;  also  let  t  and  0  de- 
note the  polar  co-ordinates  of  P,  the  pole  being 
at  the  origin  A,  and  AX  being  the  initial  line. 
Then,  as  in  Art.  26,       x=r  cos.  0, 
and  2/=^  sin.  0. 

Substituting  these  values  in  the  equation 

y—mx^ 
we  have  r  sin.  0=mr  cos.  6; 

therefore  tang.  0=m; 


THE   STEAIGUT   LINE.  61 

that  is,  0=a.  constant, 

which  is  the  polar  equation  to  a  straight  line  passing  through 

the  pole. 

Examjples,  Draw  the  straight  lines  represented  by  the  equa- 
tions 


1.  ^cos.r^-^'l^l. 

4..=| 

2.  7'cos.r0-^]=4. 

5.  e=|. 

a  ^cos.r^-^'j^s. 

6.  0==O. 

^  '     51.  The  following  examples  are  designed  to  show  how  the 
^\  preceding  principles  may  be  applied  to  the  solution  of  geomet- 
l  rical  problems. 

To  determine  whether  the  perpendiculars  draivn  from  the 
vertices  of  a  triangle  to  the  opposite  sides  meet  in  a  point. 

Let  ABC  be  any  triangle,  and  let  AE,  BF,  CD  ^    ^ 
be  perpendiculars  from  A,  B,  and  C  upon  the  op- 
posite sides. 

Let  A  be  the  origin  of  co-ordinates  ;  let  AB  be 
the  axis  of  abscissas,  and  AY,  perpendicular  to  -aTd         B)>  „  o 
AB,  the  axis  of  ordinates.     Let  the  co-ordinates  of  C  be  x^y^, 
and  those  of  B  be  x^,  0. 

Is"ow  if  the  abscissa  of  the  point  where  AE  and  BF  intersect 
is  equal  to  AD,  the  intersection  of  these  lines  must  be  on  CD. 
Since  each  of  these  lines  passes  through  a  given  point  and  is 
perpendicular  to  a  given  line,  its  equation  will  be  given  by  Art. 
45 ;  but  we  must  first  find  the  equations  to  the  lines  AC,  BC, 
to  whicli  they  are  perpendicular. 

Since  AC  passes  through  the  origin  and  the  given  point  C, 
its  equation  is  (Art.  40) 

y=%^;  (1) 

and  since  BF  passes  through  a  given  point  'B{x^^  0),  and  is 
perpendicular  to  (1),  its  equation  is  (Art.  45) 


62  ANALYTICAL   GEOMETRY. 

Also,  since  BC  passes  through  the  point  B(a?2)  0)  and  the 
point  C(a?i3/i),  its  equation  is  (Art.  40) 

and  since  AE  passes  through  the  origin  (0, 0),  and  is  perpendic- 
ular to  (3),  its  equation  is 

■  Vi  ^  ^ 

At  the  point  where  (2)  and  (4)  intersect,  their  ordinates  must 
be  identical.     Hence  we  may  equate  their  values,  and  we  have 

whence  x=x^; 

that  is,  i»,  the  abscissa  of  the  intersection  of  BF,  AE,  is  equal 
to  ajj,  the  abscissa  of  the  point  C ;  hence  the  perpendicular  CD 
passes  through  that  intersection,  and  the  three  perpendiculars 
meet  in  a  point. 

52.  To  determine  whether  the  three  perpendiculars  through 
the  middle  jpoints  of  the  sides  of  a  triangle  meet  in  appoint. 
Y  jj  Let  ABC  be  any  triangle,  and  let  D,  E,  F  be  the 

/\  middle  points  of  its  sides.     Let  P  be  the  point 

i;C^\e  where  two  of  the  perpendiculars  EP,  FP  meet ; 
^  \  now  if  the  abscissa  of  P  is  equal  to  AD,  the  inter- 
-^  »  B  section  of  the  lines  EP,  FP  must  be  in  the  per- 
pendicular drawn  from  D. 

Eepresent  the  point  C  by  {nc^y-^),  and  the  point  B  by  {x^,  0). 

X     u 
The  co-ordinates  of  F  are  -^,  ^  (Art.  24),  and  the  co-ordi- 
nates of  E  are  x,-\-x^      ^y, 

2      ^  ^  2* 
ITow  the  equation  to  AC,  passing  through  the  origin  and  the 
point  0.,^.,  is  ^j_,^^.  ^^^        : 


THE    STEAIGIIT   LINE.  63 

(ijc    ly  \ 
-^,  -~  I,  and 

,_|=_J(.-|).  (2) 

The  equation  to  BO,  passing  through  the  point  {x^,  0)  and  {x^y^, 
is  .  y=-^x-x,),  (3) 

aiiu  lb  jjerjjeiiuiijuiar  lu  \^o;,  is 

At  the  point  where  (2)  and  (4)  intersect,  their  ordinates  must 
be  identical ;  and  equating  their  values,  we  have 


\       '^\        '^2/ 

tt/j-r^c/g 

'-   y.  K' 

■     2 

^2 

X-  ^  , 

which  gives 

that  is,  a?,  the  abscissa  of  the  intersection  of  EP  and  FP,  is  equal 

X 

to  -^j  which  is  the  abscissa  of  the  point  D ;  hence  the  perpen- 
dicular from  D  passes  through  that  intersection,  and  the  three 
perpendiculars  meet  in  a  point. 


64  ANALYTICAL   GEOMETKT. 


SECTION  III. 

TRANSFORMATION   OF   CO-ORDINATES. 

53.  When  a  line  is  represented  by  an  equation  witli  refer- 
ence to  any  system  of  axes,  we  can  always  transform  that  equa- 
tion into  another  which  shall  equally  represent  the  line,  but 
with  reference  to  a  new  system  of  axes  chosen  at  pleasure. 
This  is  called  the  transformation  of  co-ordinates,  and  may  con- 
sist either  in  altering  the  relative  position  of  the  axes  without 
changing  the  origin;  or  changing  the  origin  without  disturb- 
ing the  relative  position  of  the  axes ;  or  we  may  change  both 
the  direction  of  the  axes  and  the  position  of  the  origin. 

54.  To  change  the  origin  from  onejpointto  another  without 
altering  the  direction  of  the  axes.  ■  ■' 

Let  AX,  AY  be  the  primitive  axes,  and 
let  A'X^,  A'Y'  be  the  new  axes,  respective- 
ly parallel  to  the  preceding. 

Let  AB,  A'B,  the  co-ordinates  of  the  new 
origin  referred  to  the  old  axes,  be  repre- 
A  B  M  ^  sented  by  «  and  h;  let  the  co-ordinates  of 
any  point  P  referred  to  the  primitive  axes  be  x  and  y,  and  the 
co-ordinates  of  the  same  point  referred  to  the  new  axes  be  x' 
and  y\     Then  we  shall  have 

AM=AB+BM=AB+A'M', 
or  x=a-\-x\ 

Also,  PM=:MM'-l-PM^=:BA'-t-PM', 

or  y=h+y'. 

Hence,  to  find  the  equation  to  any  line  when  the  origin  is 
changed,  the  new  axes  remaining  parallel  to  the  old,  we  must 
substitute  in  the  equation  to  the  line,  a-\-x'  for  a?,  and  h-\-y' 
for  y. 


TRANSFOEMATION   OF   CO-OEDINATES.  "      65 

These  formulas  are  equally  true  for  rectangular  and  oblique 
co-ordinates. 

Ex.  1.  Find  what  the  equation  2x-i-3y=8  becomes  when  the 
origin  is  transferred  to  a  point  whose  co-ordinates  are  a=B, 
b=l.  Ans.2x'-\-Sy'z=^l, 

Ex.2.  Find  what  the  equation  y-\-2x=—6  becomes  when 
the  origin  is  changed  to  the  point  (2, 1). 

Ans.y'  +  2x'  =  —10. 

Ex.  3.  Find  what  the  equation  y—Sx—7  becomes  when  the 
origin  is  changed  to  the  point  (—2,  —3).  Ans.  y'  =  Zx'—10, 

Ex.4.  Find  what  the  equation  2/"  +  %— 4a? +  8  =  0  becomes 
when  the  origin  is  changed  to  the  point  (1,  —2). 

Ans.  y''^—^x'. 

66.  To  change  the  direction  of  the  axes  without  changing 
the  origin^  'both  systems  being  rectangular. 

Let  AX,  AY  be  the  primitive  axes,  and 
AX',  AY'  be  the  new  axes,  both  systems 
being  rectangular.  Let  P  be  any  point ; 
a?,  y  its  co-ordinates  referred  to  the  old 
axes ;  x\  y'  its  co-ordinates  referred  to 
the  new  axes.  Denote  the  angle  XAX' 
by  0.  Through  P  draw  PR  parallel  to  AY,  and  PM  parallel 
to  AY'.  From  M  draw  MN  parallel  to  AY,  and  MQ  parallel 
to  AX. 

Then  i»=AR=A:^r-NR=AN-MQ. 

Also  AN=AM  COS.  XAX'^aj'  cos.  0, 

and  MQ=PMsin.MPQ=2/'6in.  0. 

Hence  x—x'  cos.  Q—y'  sin.  0. 

Also  2/=P^=QI^  +  PQ=M]S"+PQ. 

But  MN= AM  sin.  MAX=a;'  sin.  0, 

and  PQ = PM  cos.  MPQ = y'  cos.  0. 

Hence  V—^'  sin.  0+y'  cos.  0. 

Hence,  to  find  the  equation  to  any  line  when  referred  to 
the  new  axes,  we  must  substitute  in  the  equation  to  the  line, 
x'  cos.  ^—y'  sin.  0  for  «j,  and  x'  sin.  ^■\-y'  cos.  0  for  y. 


66  ANALYTICAL   GEOMETRY. 

Ex.1.  Find  what  the  equation  x-\-y=10  becomes  when  the 
axes  are  moved  through  an  angle  of  45°. 

mte,  sin.  45°  =  cos.  45° =-^. 

Here  i»=-^2-|-V2, 

x'  v' 

By  substitution,  the  given  equation  becomes  a;'=5  V2  A'iis. 

Ex.  2.  Find  what  the  equation  y=zZx—6  becomes  when  the 
axes  are  moved  through  an  angle  of  45°. 

Ans,  2y' =x' —Zy/'2,. 

Ex.  3.  Find  what  the  equation  y''—x^=6  becomes  when  the 
axes  are  moved  through  an  angle  of  45°.  Ans,  x'y'  =  Z. 

Ex.4.  Find  what  the  equation  ^+^=1  becomes  when  the 
axes  are  moved  through  an  angle  of  45°. 

66.  To  transform  an  equation  from  rectangular  to  ohlique 

co-ordinates. 

Let  AX,  AY  be  the  primitive  axes,  and 
AX',  AY'  be  the  new  axes.  Let  P  be  any 
point ;  a?,  y  its  co-ordinates  referred  to  the 
old  axes ;  x\  y'  its  co-ordinates  referred 
to  the  new  axes.  Through  P  draw  PR 
parallel  to  AY,  and  PM  parallel  to  AY'. 

Draw  also  MK  parallel  to  AY,  and  MQ  parallel  to  AX.    De- 
note the  angle  XAX'  by  a,  and  the  angle  XAY'  by  /3. 
Then  «=AE=AN+NE=AN+MQ. 

But  AN= AM  COS.  XAX'=aj'  cos.  a, 

and  MQ = PM  cos.  PMQ = y'  cos.  j3. 

Hence  x—x'  cos.  a^-y'  cos.  j3. 

Also  y=:PE=QR+PQ=MK4-PQ. 

But  MX = AM  sin.  X AX' = x'  sin.  a, 

and  PQ=PM  sin.  PMQ=?/'  sin.  /3. 

Hence  2^=0?'  sin.  a -f  2/'  sin.  /3. 


TEANSFORMATION   OF   CO-OEDINATES.  67 

Hence,  if  we  wish  to  pass  from  rectangular  to  oblique  axes,  we 
must  substitute  in  the  equation  to  the  line,  x'  cos.  a +2/'  cos.  /3 
for  a?,  and  x'  sin.  a^y'  sin.  /3  for  y. 

If  the  origin  be  changed  at  the  same  time  to  a  point  whose 
co-ordinates  referred  to  the  primitive  system  are  m  and  n^  these 
equations  will  become 

x=m-\rx'  cos.  a-\-y'  cos. /3. 
y=n-\-x'  sin.  a-\-y'  sin.  /3. 

In  the  following  examples  the  origin  and  the  axis  of  X  are 
supposed  to  remain  unchanged. 

Ex.  1.  Transform  the  equation  y=4:—x  from  rectangular  to 
oblique  co-ordinates,  the  new  axes  being  inclined  to  one  anoth- 
er at  an  angle  of  45°.  Ans.  x'  -\-yV^—^. 

Ex.2.  Transform  the  equation  y=3a?  from  rectangular  to 
oblique  co-ordinates,  the  new  axes  being  inclined  to  one  anoth- 
er at  an  angle  of  45°.  Ans.  Sx'+y\/2=0, 

Ex.  3.  Transform  the  equation  y=4t—x  from  rectangular  to 
oblique  co-ordinates,  the  new.  axes  being  inclined  to  one  anoth- 
er at  an  angle  of  60°.  Ans.  y\V^  + 1)  +  ^^'  =  8. 

Ex.  4.  Transform  the  equation  2x=Sy-\-6  from  rectangular 
to  oblique  co-ordinates,  the  new  axes  being  inclined  to  one  an- 
other at  an  angle  of  60°.  Ans.  2x'  +  y'{l-W^)  =  ^- 

57.  To  transform  an  equation  from  rectangular  to  polar 
co-ordinates. 

Let  AX,  AY  be  the  rectangular  axes ;  let 
B  be  the  pole  ;  and  let  BD,  the  initial  line, 
be  parallel  to  AX. 

Let  P  be  any  point ;  a?,  y  its  co-ordinates 
referred  to  the  rectangular  axes  ;  |0,  0  its  po- 
lar co-ordinates.     Draw  PM,  BC  parallel  to 
AY,  and  let  «,  h  be  the  co-ordinates  of  B  referred  to  the  prim- 
itive axes. 

ISTow  AM=AC-^CM=AC-|-BD. 

But  BD=BP  COS.  PBD=|o  cos.  Q, 

Hence  x=:a-\-p  cos.  Q. 


A.     C  M 


68  ANALYTICAL   GEOMETEY. 

Also  PM=DM+PD=BC+PD. 

But  PD=BP  sin.  PBD=^  sin.  0. 

Hence  y=b+p  sin.  6. 

Hence,  to  transform  the  equation  to  any  line  from  rectangular 

to  polar  co-ordinates,  we  must  substitute  in  the  equation  to  the 

line,  a+p  cos.  6  for  x,  and  b+p  sin.  0  for  y. 

In  the  following  examples  the  pole  is  supposed  to  coincide 
with  the  origin,  and  the  initial  line  with  the  axis  of  X. 

Ex.  1.  Transform  the  equation  x'^+y^  =  9  from  rectangular  to 
polar  co-ordinates.  Ans.  jo^cos.  ^6 + sin.  ^6)  =  9,  or  p  =  S. 

Ex.  2.  Transform  the  equation  xy=4:  from  rectangular  to 
polar  co-ordinates. 

mte.  Sin.  26=2  sin.  0  cos.  0  (Trig.,  Art.  73). 

Ans.  p"^  sin.  20=8. 

Ex.  3.  Transform  the  equation  x'^+y'^=7nx  from  rectangular 
to  polar  co-ordinates.  Ans.  p  =  7n  cos.  0. 

Ex.  4.  Transform  the  equation  x^~y'^  =  S  from  rectangular 
to  polar  co-ordinates. 

JVote.  Cos.  20 = COS.  '0  -  sin.  '0  (Trig.,  Art.  73). 

Ans.  p""  COS.  20 =S. 

58.  To  transform  an  equation  from  oblique  to  rectangular 
axes,  find  the  values  of  x'  and  y^  from  the  formulas  of  Art.  66. 

To  transform  an  equation  from  polar  to  rectangular  co-ordi- 
nates, deduce  the  values  of  p  and  0  from  the  equations  of  Art. 
57.     These  values  are 

y — b 
and  tang.  0=^^ . 


THE    CIKCLE. 


69 


SECTION  IV. 

THE   CIRCLE. 

59.  Definition.  A  circle  is  a  plane  figure  bounded  by  a 
line,  all  the  points  of  which  are  equally  distant  from  a  point 
within  called  the  centre.  The  line  which  bounds  the  circle  is 
called  its  circumference.  A  radius  of  a  circle  is  a  straight 
line  drawn  from  the  centre  of  the  circle  to  the  circumference. 

60.  To  find  the  equation  to  a  circle  referred  to  rectangular 
axes  when  the  origin  of  co-ordinates  is  at  the  centre. 

Let  A  be  the  centre  of  the  circle,  and 
P  any  point  on  its  circumference.  Let 
r  be  the  radius  of  the  circle,  and  x,  y  the 
co-ordinates  of  P.  Then,  by  Geom.,  Bk. 
iy.,Pr.ll, 

AB=4-BF=AF; 
or,     ,         .     x^-^f-r"^ 
which  is  the  equation  required. 

61.  Points  of  intersection  with  the  axes.  If  we  wish  to  de- 
termine the  points  where  the  curve  cuts  the  axis  of  X,  we  must 
put  2/=0, 

for  this  is  the  property  of  all  points  situated  on  the  axis  of  ab- 
scissas.    On  this  supposition,  we  have 

X=  ±7*, 

which  shows  that  the  curve  cuts  the  axis  of  abscissas  in  two 
points  on  different  sides  of  the  origin,  and  at  a  distance  from 
it  equal  to  the  radius  of  the  circle. 

To  determine  the  points  where  the  curve  cuts  the  axis  of  or- 
dinates,  we  make  "a; =0,  and  we  find 

y^zhr, 


70 


ANALYTICAL   GEOMETRY. 


which  shows  that  the  curve  cuts  the  axis  of  ordinates  in  two 
points  on  different  sides  of  the  origin,  and  at  a  distance  from  it 
equal  to  the  radius  of  the  circle. 

62.  Curve  traced  through  intermediate  j^oints.  If  we  wish 
to  trace  the  curve  through  the  intermediate  points,  we  reduce 
the  equation  to  the  form 

y—dz^r'  —  x', 
from  which  we  may  compute  the  value  of  y  corresponding  to 
any  assumed  value  of  x. 

Example.  Trace  the  curve  whose  equation  is  x'-^-y^'^lOO. 
By  assuming  for  x  different  values  from  0  to  11,  etc.,  we  ob- 
tain the  corresponding  values  of  y  as  given  below. 
When£c=0, 2/=:±10. 
aj=l,  2/zz:±9.95. 
iz;=r2, 2/=±9.80. 


x=4:,y=i 
x=^,y= 


9.16. 


Wheu 

a?= 

6, 

2/= 

:±8.00. 

x= 

■■  ^, 

2/= 

:±7.14. 

X- 

■  8, 

y^ 

=  ±6.00. 

X  — 

:     9, 

y^ 

:±4.36. 

X- 

:10, 

y^ 

:±0.00. 

X- 

:11, 

y  is  imaginary. 

"When  x—0,y  will  equal  ±10,  which 
gives  two  points,  a  and  a\  one  above 
and  the  other  below  the  axis  of  X. 
When  x=l,  3/=  ±9.95,  which  gives  the 
points  5  and  ^'.  Whenaj=:2,  ?/=±9.80, 
which  gives  the  points  c  and  c',  etc.  If 
we  suppose  x  greater  than  10,  the  value 
of  y  will  be  imaginary,  which  shows  that 

the  curve  does  not  extend  from  the  centre  beyond  the  value 

a;=:10. 

If  ic  is  negative,  we  shall  in  like  manner  obtain  points  in  the 

third  and  fourth  quadrants,  and  the  curve  will  not  extend  to 

the  left  beyond  the  value  a?=  — 10. 

Since  every  value  of  x  furnishes  two  equal  values  of  y  with 

contrary  signs,  it  follows  that  the  curve  is  symmetrical  above 

and  below  the  axis  of  X. 


THE   CIECKE. 


71 


63.  To  find  the  equation  to  a  circle  when  the  origin  is  on 
the  circumference,  and  the  axis  qfS^  passes  through  the  centre. 

Let  the  origin  of  co-ordinates  be  at  A,  a 
point  on  the  circumference  of  the  circle,  and 
let  the  axis  of  X  pass  through  the  centre. 
Let  r  be  the  radius  of  the  circle,  and  let  x,  y 
be  the  co-ordinates  of  P,  any  point  on  the 
circumference.  Then  CB  will  be  represent- 
ed by  x—r. 

Now  CB^-f-BF  =  CP^ 

or  {x—ry-{-y''=r% 

whence  2/^  =  2rx — sf, 

which  is  the  equation  required. 


64.  Points  of  intersection  with  the  axes.  If  we  wish  to  de- 
termine where  the  curve  cuts  the  axis  of  X,  we  make  y=Oy 
and  we  find  x{^r—x)  =  0. 

This  equation  is  satisfied  by  supposing  x=0,  or  2/*— aj=0, 
from  the  last  of  which  equations  we  find  x=z2r.  The  curve, 
therefore,  cuts  the  axis  of  abscissas  in  two  points,  one  at  the 
origin,  and  the  other  at  a  distance  from  it  equal  to  '^r. 

To  determine  where  the  curve  meets  the  axis  of  ordinates, 
we  make  x—0,  which  gives 

which  shows  that  the  curve  meets  the  axis  of  ordinates  in  but 
one  point,  viz.,  the  origin. 


65.  Curve  traced  through  intermediate  points.  In  order  to 
trace  the  curve  through  intermediate  points,  we  reduce  the 
equation  to  the  form 

y=±^/'2irx--x^, 
from  which  we  may  compute  the  value  of  y  corresponding  to 
any  assumed  value  of  x,  as  in  Art.  62. 

Ex.  1.  Trace  the  curve  whose  equation  is  y'^  =  10x—x'^. 

By  assuming  for  x  different  values  from  0  to  11,  etc.,  we  ob- 
tain the  corresponding  values  of  y  as  given  on  the  next  page. 


72 


ANALYTICAL   GEOMETRY. 


When  x=0,  y=0. 
x=l,y=S, 

x=2,  y=:4:. 

x=S,  y=-4:.58. 

X=z4:,  2/=:  4.90. 

x=5,  y=5. 


Whenir=  6, 2/=4.90. 

x=  7,y=4:.68. 

x=  8,y=4:, 

x=  9, 2/=3. 

aj=10,  y=0. 

x=ll,  yis  imaginary. 
These  values  may  be  represented  by  a  figure  as  in  Art.  62. 
Ex.  2.  Trace  the  circle  x''-\-y''  =  10y. 
Ex.  3.  Trace  the  circle  x''-{-y'=  —10a?. 

66.  To  find  the  equation  to  the  circle  referred  to  any  rect- 
angular axes. 

Let  C  be  the  centre  of  the  circle,  and 
P  any  point  on  its  circumference.  Let  r 
be  the  radius  of  the  circle ;  a  and  h  the 
co-ordinates  of  C ;  x^y  the  co-ordinates 
of  P.  Erom  C  and  P  draw  lines  perpen- 
dicular to  AX,  and  draw  CD  parallel  to 
-X  AX.     Then 

CD=-fDP=:CP; 

that  i  s,  {x—df-\-  {y — h^ = r^, 

which  is  the  equation  required. 

67.  Varieties  in  the  equation  to  the  circle.  If  in  the  equa- 
tion {x—ay-\-{y—by=r^  we  suppose  a=:0  and  h  =  0,  the  centre 
of  the  circle  becomes  the  origin  of  co-ordinates,  and  the  equa- 
tion becomes 

x'-\-y''=r''  (as  in  Art.  60). 

If  we  suppose  a=r  and  ^=0,  the  axis  of  X  becomes  a  diam- 
eter, and  the  origin  is  at  its  extremity,  and  the  equation  be- 
comes {x—ry+y'':=r'', 
whence                  2^''  =  2ric— a?"  (as  in  Art.  63). 

If  we  suppose  a=:0  and  b=r,  the  axis  of  Y  becomes  a  diam- 
eter, and  the  origin  is  at  its  extremity,  and  the  equation  be- 
comes ^'^ + (y — ^y = ^^ 

whence  x''  =  2ry--y\ 


THE    CIRCLE.  73 

68.  General  equation  to  the  circle.  Expanding  the  general 
equation  to  the  circle  referred  to  rectangular  axes,  we  have 

'  x^^y'-'^ax-Uy^a^-\-V-r^^O', 
and  hence  it  appears  that  the  general  equation  to  the  circle  is 
of  the  form 

where  A,  B,  and  C  are  constant  quantities,  any  one  or  more  of 
which  in  particular  cases  may  be  equal  to  zero.     The  equation 

Aaj''+A/+Baj+C2/+D  =  0 
may  be  reduced  to  this  form  by  dividing  by  A,  and  is  therefore 
the  most  general  form  that  the  equation  can  assume  when  the 
co-ordinates  are  rectangular. 

69.  To  determine  the  circle  represented  hy  an  equation.  If 
we  can  reduce  an  equation  to  the  form 

x'JrfJrAx-^By-^C  =  0, 

A  2|T>2 

we  may  determine  the  circle  it  represents ;  for,  adding  — j — 

to  both  sides  of  the  equation,  and  transposing  C,  we  have 
/      AV    /      BV    A^+B»    ^ 

By  comparing  this  equation  with  that  of  Art.  Q>Q,  we  perceive 
that  it  represents  a  circle,  the  co-ordinates  of  whose  centre  are 

—K^—K^  and  whose  radius  is 

(^'-c)*ori(A'+B'-4C)i 

If  A''+B''<4C,  the  radius  becomes  imaginary,  and  the  equa- 
tion can  represent  no  real  curve. 

Ex.  1.  Determine  the  co-ordinates  of  the  centre,  and  the  ra- 
dius of  the  circle  denoted  by  the  equation  aj'-f  2/^+4a?— 8y— 5 
=0. 

This  equation  may  be  reduced  to  the  form 

Hence  the  co-ordinates  of  the  centre  are  —2, 4,  and  the  ra- 
dius is  5.  I  . 

D 


74  ANALYTICAL   GEOMETRY. 

Ex.  2.  Determine  the  co-ordinates  of  the  centre  and  the  ra^ 
dius  of  the  circle  denoted  by  the  equation  x'-\-7/-i-4:y—4^x—l 
=  0.  Ans.  Co-ordinates  2,  —2,  radius  3. 

Ex.  3.  Determine  the  co-ordinates  of  the  centre  and  the  ra- 
dius oithe  circle  denoted  by  the  equation  x'^+y''-\-6x—4:y—3Q 
=  0.  c  Ans.  Co-ordinates  —  3, 2,  radius  7. 

Ex.  4.  Detemine  the  co-ordinates  of  the  centre  and  the  ra- 
dius of  the  circle  denoted  by  the  equation  x'^-\-y'^~3x—4:y-{-4: 
—  0.  Ans.  Co-ordinates  |-,  2,  radius  -f. 

Ex.  6.  Determine  the  co-ordinates  of  the  centre  and  the  ra- 
dius of  the  circle  denoted  by  the  equation  x''-^y'^  —  2a(x—y) 

Ans.  Co-ordinates  a,  —a,  radius  (2a^-\-c^y. 

Ex.  6.  Find  the  equation  to  the  circle  whose  radius  is  9,  and 
co-ordinates  of  the  centre  —1, 5. 

Ex.  7.  Find  the  equation  to  the  circle  whbse  radius  is  6a,  and 
co-ordinates  of  the  centre  8a,  4:a. 

70.  To  find  ihejpolar  eqiiation  to  a  circle  when  the  origin  is 
on  the  circumference,  and  the  initial  line  is  a  diameter. 

Let  A  be  the  pole  situated  on  the 
circumference  of  the  circle ;  let  AX, 
passing  through  the  centre,  be  the  ini- 
tial line,  and  let  P  be  any  point  on  the 
circumference.  Let  r  be  the  radius  of 
the  circle,  and  let  p  and  Q  be  the  polar 
co-ordinates  of  P. 
The  equation  of  the  circle  referred  to  rectangular  axes  (Art. 
63)  is  y''  =  2rx-x\ 

To  transform  this  equation  from  rectangular  to  polar  co-or- 
dinates (Art.  59),  we  must  substitute  for  x,  p  cos.  6 ;  and  for 
y,  p  sin.  e. 

Making  this  substitution,  we  obtain 

p""  sin.  ''0=2rp  cos.  0—p''  cos.  ""0; 
or,  by  transposition, 

p\&m.'e-\-cos.'e)=^2rp  cos.  0. 


THE   CIRCLE.  75 

But  sin.  ^d-{-GOS.  ""Q  is  equal  to  unity. 

Hence,  dividing  by  p,  we  obtain 

p=2r  COS.  0, 
which  is  the  polar  equation  of  the  circle. 

71.  Points  of  the  circle  determined.  "When  0=0,  cos.  0=1, 
and  we  have 

|0  =  2r=AB. 

As  Q  increases  from  0  to  90°,  the  radius  vector  determines 
all  the  points  in  the  semi-circumference  BPA ;  and  w^hen  0 
=  90°,  COS.  0=0,  and  p  becomes  zero. 

From  0=90°  to  0=180°  the  radius  vector  is  negative,  and  is 
measured  into  the  fourth  quadrant,  determining  all  the  points 
in  the  semi-circumference  below  the  axis  of  abscissas.  From 
0=180°  to  0=360°  the  circumference  is  described  a  second 
time. 

Ex.  1.  The  polar  co-ordinates  of  P  are  /o  =  10, 0=45° ;  deter- 
mine the  radius  of  the  circle. 

Ex.  2.  The  radius  of  a  circle  is  5  inches,  and  jo  =  8  inches ; 
determine  the  value  of  0. 

Ex.  3.  The  radius  of  a  circle  is  5  inches,  and  0=60° ;  deter- 
mine the  radius  vector. 

72.  Definition.  Let  two  points  be  taken  on  a  curve,  and  a 
secant  line  be  drawn  through  them ;  let  the  first  point  remain 
fixed,  while  the  second  point  moves  on  the  curve  toward  the 
first  until  it  coincides  with  it ;  when  the  two  points  coincide, 
the  secant  line  becomes  a  tangent  to  the  curve. 

Suppose  a  straight  line  MP  to  intersect 

a  curve  in  two  points,  M  and  P,  and  let  ^        >^^<^ ■T'' 

the  line  turn  about  the  fixed  point  P  until 
it  comes  into  the  position  PM'.  The  sec- 
ond point  of  intersection,  which  at  first  was  on  the  left  of  P,  is 
now  found  on  the  right  of  P ;  hence,  in  the  movement  of  the 
straight  line  from  the  position  MP  to  the  position  PM^,  there 
must  have  been  one  position  in  which  the  point  M  coincided 


& 


76  ANALYTICAL   GEOMETEY. 

with  p.     In  this  position,  represented  by  the  line  TT',  the  line 
is  said  to  be  a  tangent  to  the  curve. 

This  definition  of  a  tangent  suggests  a  method  of  finding  its 
equation  which  is  applicable  to  all  curves. 

\  73.  To  find  the  equation  to  the  tangent  at  any  jpoint  of  a 
circle. 

Let  the  equation  to  the  circle  be  x^-\-y''=T''. 

Let  x\  y'  be  the  co-ordinates  of  the  point  on  the  circle  at 
which  the  tangent  is  drawn,  and  x'\  y"  the  co-ordinates  of  an 
adjacent  point  on  the  circle.  The  equation  to  the  secant  line 
passing  through  the  points  x',  y'  and  x'\  y"  (Art.  40)  is 

2/-2/ =J^'(^-^>  (1) 

Kow,  since  the  points  x\  y'  and  x'\  y"  are  both  on  the  cir- 
cumference of  the  circle,  we  must  have 

or  y"^-^j'^=^x''-x''%  r    -  [V     -A    ^  / 

,  y"—y'        x"-\-x'  ■ 

whence  —, — —,  =  —~Tr, — i- 

X  —X         y  +y 

Substituting  this  value  in  equation  (1),  we  obtain 

which  is  the  equation  to  the  secant  line  passing  through  the 
two  given  points. 

Now  when  the  point  x',  y'  coincides  wdth  the  point  x",  y" ^  we 
have  x'=x'\  and  y' —y" ;  hence  equation  (2)  becomes 

x' 

which  is  the  equation  to  the  tangent  at  the  point  x\  y\  where  x 
and  y  are  the  co-ordinates  of  any  point  of  the  tangent  line. 
Clearing  of  fractions  and  transposing,  w^e  obtain 
xx'-{-yy'=x'^-^ij'% 
or  xx'  -\-yy'=:r'^, 

which  is  the  simplest  form  of  the  equation  to  the  tangent  line. 


THE   CIECLE. 


Y7 


■   74.  Points  where  the  tangent  cuts  the  axes.     To  determine 
the  point  in  which  the  tangent  inter- 
sects the  axis  of  X,  we  make  2/=0, 
which  gives 


or 


x———Mj^ 


since  x  is  AC  when  2/=0. 

To  determine  the  point  in  which, 
the  tangent  intersects  the  axis  of  Y,  we  make  x—^^  which  gives 


or 


y         y 


Ex.  1.  On  a  circle  whose  radius  is  6  inches,  a  tangent  line  is 
drawn  through  the  point  whose  ordinate  is  4  inches ;  determine 
where  the  tangent  line  meets  the  two  axes ;  also  the  angle 
which  the  tangent  line  makes  with  the  axis  of  X. 

Ex.  2.  Find  the  point  on  the  circumference  of  a  circle  whose 
radius  is  5  inches,  from  which,  if  a  radius  and  a  tangent  line 
be  drawn,  they  will  form  with  the  axis  of  X  a  triangle  whose 
area  is  35  inches.    >.    ^   ^^  _    .„1  I.. 

75.  To  find  the  length  of  the  tangent  draion  to  the  circle 
from  a  given  jpoint. 

Let  P  be  a  point  without  the  circle  from 
which  a  tangent  line  PM  is  drawn.  Draw 
the  radius  AM,  and  join  AP.  Let  the  co-or- 
dinates of  P  be  a?,  y.     Then  we  have 

PM^  =  AP-AMl 
But  AP=a;'-f2/''(Art.l9). 

Hence  VM.={x'-\-y'-r')^, 

which  denotes  the  length  of  the  tangent  line  from  the  point  a?,  y. 
If  x^-\-y'^yr^,  or  the  point  P  be  without  the  circle,  the  tan- 
gent PM  will  be  real ;  if  o^-^y^—r''^  or  the  point  P  be  on  the 
circle,  the  length  of  the  tangent  becomes  zero  ;  if  x^-\-y''<r'',  or 


78  ANALYTICAL   GEOMETRY. 

the  point  P  be  within  the  circle,  tlie  tangent  is  imaginary ;  but 
the  quantity  r^—x^—y^  represent^- the  product  of  the  segments 
of  the  chord  drawn  through  P.  t^'t>^^  /w  ^ 

Ex.  1.  Find  the  length  of  the  tangent  drawn  from  the  point 
—7,  +5,  to  a  circle  whose  radius  is  4. 

Ex.  2.  Find  the  length  of  the  tangent  drawn  from  the  point 
~3,  —  6,  to  a  circle  whose  radius  is  5. 


Qm 


76.  Definition.  The  normal  at  any  point  of  a  curve  is  a 
straight  line  drawn  through  that  point  perpendicular  to  the 
tangent  to  the  curve  at  that  point. 

77.  To  find  the  equation  to  the  normal  at  any  jpoint  of  a 
circle. 

Let  the  equation  to  the  circle  be  x'^-\-y'^=r^,  and  let  x\  y'  be 
the  co-ordinates  of  the  point  on  the  circle  through  which  the 
normal  is  drawn. 

We  have  found  (Art.  73,  Eq.  3)  that  the  equation  to  the  tan- 
gent at  the  point  x' ^  y'  is 

x'  " 

where  —  —  denotes  the  tangent  of  the  angle  which  the  tangent 

line  makes  with  the  axis  of  X.  Hence  (Art.  46)  the  equation 
to  the  normal  will  be 

which,  after  reduction,  becomes 

y' 

^     x'  ' 
and  this  is  the  equation  to  the  normal  passing  through  the  giv- 
en point. 

y' 

We  have  found  (Art.  40)  that  y—~,x  is  the  equation  to  a 

yb 

straight  line  passing  through  the  origin  and  through  a  given 
point ;  hence  the  normal  at  any  point  of  a  circle  passes  through 
the.  centre. 


THE   CIECLE.  79 

78,  To  determine  the  co-ordinates  of  the  poiiits  of  intersec- 
tion of  a  straight  line  with  a  circle. 

Let  the  equation  to  the  circle  be 

x^-{-i/=r%  (1) 

and  the  equation  to  the  straiglit  line  be 

y^mx-\-c.  (2) 

Since  the  co-ordinates  of  every  point  on  a  line  must  satisfy 
its  equation,  the  co-ordinates  of  the  points  through  which  both 
of  the  given  lines  pass  must  satisfy  both  equations.  We  may 
therefore  regard  (1)  and  (2)  as  simultaneous  equations  contain- 
ing but  two  unknown  quantities,  and  we  may  hence  determine 
the  values  of  x  and  y.  By  substitution  in  equation  (1)  we  ob- 
tain 

x"  -f  m  V + 2cmx +&= r"^ 
or  (1 + 7)1^)^  +  '^cmx —r^— c', 

an  equation  of  the  second  degree  which  may  be  solved  by  com- 
pleting the  square.     We  thus  find 

— cm  ±  V^Yl  +  m') — c' 
X— 


and  since  x  has  two  values,  we  conclude  that  there  will  be  two 
points  of  intersection. 

If  rXl+m^)  =  c'',  the  two  values  of  x  become  equal,  and  the 
straight  line  will  touch  the  circle.  If  r^i).  -f  m^)  is  less  than  &^ 
the  straight  line  will  not  meet  the  circle. 

Ex.  1.  Find  the  co-ordinates  of  the  points  in  which  the  circle 
whose  equation  is  x^-\-y'^—1^o  is  intersected  by  the  line  whose 
equation  is  a? -i- 2/= 1.  .       j       cc=:4,  and  ?/=— 3, 

*  I  or  a?=  — 3,  and  y=^. 

Ex.  2.  Find  the  co-ordinates  of  the  points  in  which  the  circle 
whose  equation  is  oi^-^y'  —  '^'^  is  intersected  by  the  line  whose 
equation  is  i«+?/= 5.  .       (       ^c^S,  and  2/=0, 

'  I  or  a?r=0,  aud  y='^. 

Ex.  3.  Find  the  co-ordinates  of  the  points  in  which  the  circle 
whose  equation  is  x^-\-y^-=^^  is  intersected  by  the  line  whose 
equation  is  30?+?/= 25.^  .       j       x=1^  and  y—^, 

'  I  or  a?=i8,  and  y~i. 


80  ANALYTICAL   GEOMETRY. 

Ex.  4.  Find  the  points  in  which  the  line  3/=:  5a; +  2  intersects 
the  circle  9/-]-x^—4:y—13x=9. 

AnsA      «^=l>and2/=7, 
.   i  or  x=—i,  ana  y=—t- 
Ex.  5.  Find  the  points  in  which  the  line  y=dx-\-2  cuts  the 
circle  ?/''+a;''— 4a?4-4?/=T. 

79.  To  find  the  co-ordinates  of  the  jpoints  of  intersection  of 
two  circumferences. 

Let  CPP^,  DPP'  be  two  cir- 
cumferences which  intersect  in 
P  and  P'.  Let  A  and  B  be  the 
centres  of  the  circles,  r  and  r' 
their  radii,  and  let  AB,  the  dis- 
tance between  their  centres,  be 
denoted  by  d.  Assume  the  line 
AB  as  the  axis  of  X,  and  let  AY  be  drawn  perpendicular  to 
AX  for  the  axis  of  Y. 

The  equation  to  the  circle  CPP'  is 

x-'-^-y'^rK  (1) 

The  equation  to  DPP',  the  co-ordinates  of  whose  centre  are 
{d,  0)  (Art.  ^^\  is 

{x-d'f-^y''=r'\  (2) 

Since  the  co-ordinates  of  every  point  of  a  circumference 
must  satisfy  the  equation  of  the  circle^  the  co-ordinates  of  the 
points  through  which  both  circumferences  pass  must  satisfy 
both  equations.  We  may  therefore  regard  (1)  and  (2)  as  sim- 
ultaneous equations  involving  but  two  unknown  quantities,  and 
hence  we  may  determine  the  values  of  x  and  y.  Subtracting 
equation  (2)  from  equation  (1),  we  obtain 
/    2xd-d'=r''-r'% 

.                            X         r'^-r'^-^d'' 
whence  fV= ^ . 

Substituting  this  value^>^of  x  in  equation  (1),  we  have 


^=-r-?^): 


THE   CIRCLE.  81 


H 


whence        \    y=  :^-^V^d'r'-if-r"-\-dJi 

which  gives  the  ordinates  of  the  points  of  intersection  of  the 
two  circles. 

The  double  sign  of  y  shows  that  the  two  points  of  intersec- 
tion have  the  same  abscissa  AE,  but  two  ordinates  numerically 
the  same  and  with  contrary  signs.  Hence,  when  two  circum- 
ferences cut  each  otlier,  the  line  joining  their  centres  is  perpen- 
dicular to  the  common  chord,  and  divides  it  into  two  equal 
parts. 

Ex.  1.  Find  the  co-ordinates  of  tlie  points  of  intersection  of 
the  two  circumferences 

aj^+?/''  =  25,  and  x'-\-f+14:x=-lZ. 

Ans.  x= -2.114c',  2/=  ±4.199. 
Ex.  2.  Find  the  co-ordinates  of  the  points  of  intersection  of 
the  two  circumferences  x^-\-y'^=(S,  and  x^ -\-y'^ —^x=  —  S. 

Ans.x=1.1^]  y  =±1.714:. 
Ex.  3.  Find  the  co-ordinates  of  the  points  of  intersection  of 
tlie  two  circumferences  ,       , 

x'^+y''—2x—4y=l,  and  x''-\-y^—4:X—()y=—6. 

80.  To  find  the  equation  to  the  straight  line  which  passes 
through  the  points  of  intersection  of  two  circles  which  cut  each 
other. 

Let  the  equations  of  the  two  circumferences,  whose  centres 
are  at  B  and  C,  be  severally 

x'-\-y'\ax-^ly-^c=(),  (1) 

and  x^j^f-\-a'x^h'y-\-c'  =  ^\  (2) 

it  is  required  to  find  the  equation  of  the  straight  line  passing 
tlirough  the  points  P  and  P'  where  these  circumferences  inter- 
sect. 

Since  the  co-ordinates  of  the  points  P  and  P'  satisfy  each  of 
the  above  equations,  we  may  treat  them  as  simultaneous  equa- 
tions containing  two  unknown  quantities. 

Subtracting  equation  (2)  from  equation  (1),  we  have 

{a-a')x-{-q)-h')y-^c-c'  =  ^.  (3) 

D2 


82 


ANALYTICAL    GEOMP^TRY. 


Since  tliis  is  an  equation  of  the  first 
degree  between  x  and  ?/,  it  is  the  equa- 
tion of  a  straight  line  (Art.  37) ;  and 
since  it  must  be  satisfied  by  the  co- 
ordinates of  the  two  points  P  and  P', 
it  must  be  the  equation  of  the  straight 
line  DE  passing  through  those  points, 
and  is,  therefore,  the  equation  re- 
quired. 

If  we  combine  equation  (3)  with  the  equation  of  either  cir- 
cle, we  shall  obtain  the  values  of  the  co-ordinates  of  the  points 
of  intersection  as  in  Art.  79. 

In  general,  if  we  have  any  two  equations  of  curves,  and  we 
add  or  subtract  those  equations  as  in  the  process  of  elimination 
in  Algebra,  we  obtain  a  new  equation,  which  is  the  equation  of 
a  new  line  or  curve  which  passes  through  the  points  of  inter- 
section of  the  first  two  curves. 


81.  To  find  the  equation  to  a  circle  whicJi  passes  through 
three  given  points. 

We  have  found  (Art  68)  that  the  general  equation  to  the 
circle  is  x''^y''-\-Ax-\-'By^C  =  0, 

where  A,  B,  and  C  are  constant  for  a  given  circle,  but  vary  for 
different  circles ;  so  that  when  A,  B,  and  C  are  known,  the  cir- 
cle is  fully  determined. 

If  the  three  points  x'y',  x"y"^  x"'y"'  are  on  the  circumfer- 
ence of  a  circle,  the  co-ordinates  of  each  of  these  points  must 
satisfy  the  equation  of  that  circle.  If  then  we  substitute  the 
values  of  x' ^  y'  in  the  general  equation,  we  shall  obtain  an  equa- 
tion which  expresses  the  relation  between  the  coefiicients  A,  B, 
and  C.  So  also,  if  we  substitute  successively  the  values  of 
x"y"  and  x"'y"\  we  shall  obtain  two  other  equations  express- 
ing the  relations  between  the  same  coefficients.  We  shall  then 
have  three  simultaneous  equations  expressing  the  relations  be- 
tween the  three  quantities  A,  B,  and  C,  from  which  the  values 
of  these  quantities  can  be  determined. 


THE   CIRCLE.  83 

Ex.  1.  Find  the  equation  to  the  circle  which  passes  through 
the  three  points  1,  2  ;  1,  3 ;  and  2,  5  ;  also  the  co-ordinates  of 
the  centre  and  the  radius  of  the  circle. 

Substituting  these  values  successively  in  the  general  equa- 
tion of  the  circle,  we  have 

A+2B+C  +  5  =  0, 
A+3B  +  C  +  10  =  0, 
2A+5B  +  C-f29  =  0, 
from  which  we  find     A  =  —  9 ;  B  =  —  5  ;  0  =  14. 
Hence  the  equation  to  the  circle  is  x'^  +  7/—9x—6y-^14:=:0. 
Hence  the  co-ordinates  of  the  centre  are  -|,  f ;  and  the  radius 

is|V2. 

Ex.  2.  Find  the  equation  to  the  circle  which  passes  through 
die  three  points  2,  —  3 ;  3, — 4;  and  —  2,  —  1 ;  also  the  co-ordi- 
nates of  the  centre  and  the  radius  of  the  circle. 

Ans.  Eq.,  a;' + 2/'  +  8a?  -f  20?/ +31  =  0;  co-ordinates,  —  4,  — 10 ; 
radius  ^-v/SS. 

Ex.  3.  Find  the  equation  to  the  circle  which  passes  through 
the  origin  and  through  the  points  2, 3  and  3, 4 ;  also  the  co-or- 
dinates of  the  centre  and  radius  of  circle. 

Ans.  Eq.^Qi^-^y^—'^Zx^Wy—^^  co-ordinates, -y-, —^;  ra- 
dius =4V26. 

Ex.  4.  Find  the  equation  of  the  circle  which  passes  through 
the  three  points  —4,-4;  — 4,  —  2 ;  —  2,  +2 ;  also  the  co-ordi- 
nates of  the  centre  and  radius  of  circle. 

Ans.  Ec[.^Qi?-\-y^  —  ^x-\-^y—2>'^  —  ^\  co-ordinates,  3, —3;  ra- 
dius, 5  '/2. 

Ex.  5.  Find  the  equation  of  the  circle  which  passes  through 
the  points  —2,  —4;  2,  2 ;  4,  4;  also  the  co-ordinates  of  the 
centre  and  radius  of  circle. 

Ans.  ^^.,aj'+2/'— 42aj+302/4-16  =  0;  co-ordinates,  21, -15  ; 
radius  =5-v/26. 

Ex.  6.  Find  the  equation  of  the  circle  which  passes  through 
the  origin  and  cuts  off  lengths  6,  8  from  the  axes ;  also  the  co- 
ordinates of  the  centre  and  radius  of  circle. 

Ans.  Eq.^^-\-y^—^x~%y—fd'^  co-ordinates,  3, 4 ;  radius,  5. 


M 


ANALYTICAL   GEOMETEY. 


SECTION  Y. 


THE     PARABOLA. 


82.  A  parabola  is  a  plane  curve  every  point  of  which  is 
equally  distant  from  a  fixed  point  and  a  fixed  straight  line. 

The  fixed  point  is  called  i\\Q  focus  of  tlie  parabola,  and  the 
fixed  straight  line  is  called  the  directrix. 

Thus,  if  a  straight  line  BC,  and  a  point  F 
without  it  be  fixed  in  position,  and  the.  point 
P  be  supposed  to  move  in  such  a  manner 
that  PF,  its  distance  from  the  fixed  point,  is 
always  equal  to  PD,  its  perpendicular  dis- 
tance from  the  fixed  line,  the  point  P  will 
describe  a  parabola  of  w^hich  F  is  the  focus 
and  BC  the  directrix. 

83.  From  the  definition  of  a  parabola  the  curve  may  be  de- 
scribed mechanically  by  means  of  a  ruler,  a  square,  arid  a  cord. 

Let  BC  be  a  ruler  whose  edge  coincides 
with  the  directrix  of  the  parabola,  and  let 
DEG  be  a  square.     Take  a  cord  whose 
length  is  equal  to  DG,  and  attach  one  ex- 
tremity of  it  at  G  and  the  other  at  the  fo- 
cus F.     Then  slide  the  side  of  the  square 
DE  along  the  ruler  BC,  and  at  the  same 
time  keep  the  cord  continually  stretched  by 
means  of  the  point  of  a  pencil,  P,  in  contact 
with  the  square  ;  the  pencil  will  trace  out  a  portion  of  a  parab- 
ola.    For,  in  every  position  of  the  square, 
PF+PG=PD  +  PG, 
and  hence  PF=PD; 

that  is,  the  point  P  is  always  equally  distant  from  the  focus  F 
and  the  directrix  BC. 


THE   PARABOLA. 


85 


If  the  square  be  turned  over,  and  moved  on  the  other  side 
of  the  point  F,  the  other  part  of  the  same  parabola  may  be  de- 
scribed. 


84.  A  straight  line  drawn  through  the  focus  perpendicular 
to  the  directrix  is  called  the  axis  of  the  parabola.  The  vertex 
of  the  axis  is  the  point  in  wliich  it  intersects 
the  curve.  The  chord  drawn  through  the  fo- 
cus of  a  parabola  at  right  angles  to  the  axis 
is  called  the  latus  recticm. 

Thus,  in  the  figure,  BX  is  the  axis  of  the 
parabola,  A  is  the  vertex  of  the  axis,  and  LL' 
is  the  latus  rectum. 


85.  To  find  the  equation  to  the  jparalola  referred  to  rectan- 
gular axes. 

Take  the  directrix  YY'  as  the  axis  of 
ordinates,  and  BX,  drawn  perpendicular  to 
it  through  the  focus,  as  the  axis  of  abscis- 
sas.    LetBF=2^.     By  the  definition, 

'     FP=PD:=BN. 
Therefore     FP'^=B]S[^ 
or  FN^4-PN^=BN''; 

that  is,  {x  —  ^a)"-  -\-if— x", 

or  7/=4:a{x—a), 

which  is  the  equation  to  the  parabola. 

If  in  this  equation  "vve  put  y=0,  we  have  x=a,  which  shows 
that  the  curve  cuts  the  axis  at  a  point  A  which  bisects  BF. 

The  equation  will  be  simplified  if  we  put  the  origin  at  A. 
Let  aj'  =  AN;  then  x=x'-\-a;  and,  since  the  axis  of  abscissas 
remains  unchanged,  y^y'-  >^ 

By  substitution,  equation  (1)  becomes 
y"=4.ax'. 

We  may  suppress  the  accents  if  we  remember  that  the  origin 
is  now  at  A ;  thus  w^e  have 

7/=Ux,  (2) 


86  ANALYTICAL    GEOMETEY. 

which  is  the  equation  to  the  parabola  referred  to  its  vertex  as 
origin,  and  the  axis  of  the  parabola  is  the  axis  of  X. 

86.  To  trace  the  form  of  the  ^parabola  from  its  equation. 
Since  y^—^ax^  or  a?=  t— ,  x  can  not  be  negative  ;  that  is,  the 

curve  lies  wholly  on  the  positive  side  of  the  axis  of  y. 

Since  y''—^ax^  y—^  2{axY  ; 

therefore,  since  this  equation  is  unaltered  if  we  write  —y  for 
y,  to  every  point  P  on  the  curve  on  one  side  of  the  axis  of  X, 
there  corresponds  another  point  P'  on  the  other  side,  such  that 
P'N=:PN.  Hence  the  curve  is  symmetrical  with  respect  to 
the  axis  of  X. 

Again,  if  x=0,  V—^i  ai^d  has  no  other  value  ;  therefore  the 
curve  does  not  meet  either  axis  at  any  other  point  besides  the 
origin. 

Also,  the  greater  the  value  we  give  to  x^  the  greater  values  we 
get  for  y  ;  and  when  x  is  infinite,  y  is  infinite ;  hence  the  curve 
goes  off  to  an  infinite  distance  on  each  side  of  the  axis  of  X. 

87.  To  find  the  distance  ofanyjpoint  on  the  curve  from  the 
focus. 

The  distance  of  any  point  on  the  curve  from  the  focus  is 
equal  to  the  distance  of  the  same  point  from  the  directrix. 
Hence  FP=:PD=rBA4-AI^, 

or  FP=:<3^+a?. 

88.  To  find  the  length  of  the  latus  rectum. 
In  the  equation 

y'^=4cax, 
put  x=a; 

then  y'*=4ca*, 

and  2/=±2<^, 

or  the  latus  rectum  lU=z4:a  (see  figure  in  Art.  84). 

If  w^e  convert  the  equation  y'  —  ^ax  into  a  proportion,  we 
rhall  have  x'.ywy.^a; 


''l/^A^i 


THE   PARABOLA. 


87 


that  is,  the  latus  rectwm  is  a  third jprojportional  to  any  abscissa 
and  its  corresponding  ordinate. 

89.  The  squares  ofordinates  to  the  axis  are  to  each  other  as 
their  corresponding  abscissas. 

Designate  any  two  ordinates  by  y\  y'\  and  the  corresponding 
abscissas  by  x\  x" ;  then  we  shall  have 

y^'^^^ax', 


4hfK 


What  is  the 


y"'=^4:ax". 
Hence  y"" :  y'"" : :  ^ax' :  4:ax"  wx'  :x\ 

Ex.1.  The  equation  of  a  parabola  is  y^—^x. 
abscissa  corresponding  to  the  ordinate  7  ?  "^Ans.  12 J. 

Ex.  2.  The  equation  of  a  parabola  is  y^  —  \%x.     What  is  the 
ordinate  corresponding  to  the  abscissa  7  ? 

Ans.  ±Vl26. 
Ex.  3.  The  equation  of  a  parabola  is  y^  —  Vdx.     What  is  the 
ordinate  corresponding  to  the  abscissa  3  ? 

90.  To  trace  the  form  of  the  parabola  by  means  of  points. 
If  we  reduce  the  equation  of  the  parabola  to  the  form 
?/=±2'v/aa?, 
we  may  compute  the  values  of  y  corresponding  to  any  assumed 
value  of  X. 

Ex.  1.  Trace  the  curve  whose  equation  is  y^—^x. 
By  assuming  for  x  different  values  from  0  to  5,  etc.,  we  ob- 
tain the  corresponding  values  of  y  as  given  below. 
When  aj=:0,  y—^. 
«     aj=l,  2/=:±2. 
^  "      aj=2,  2/==  ±2.828. 
«     aj=3, 2/==±3.464r. 
"     a?=4, 2/=±4. 
"     aj=5, 2/=±4.472. 
The  first  point  (0, 0)  is  the  origin ;  the 
point  (1,  +2)  is  represented  by  a  in  the  fig- 
ure ;  the  point  (1,  —2)  by  a'  in  the  figure ; 
the  point  (2,  -1-2.828)  by  b;  the  point  (2,  —2.828)  by  b\  etc. 


88  ANALYTICAL    GEOMETRY. 

Ex.  2.  Trace  the  curve  whose  equation  is  y^  =  18x. 

Ex.  3.  Trace  the  curve  whose 
equation  is  x^  =  dy. 

The  curve  will  be  of  the  form 
exhibited  in  the  annexed  figure, 
and  is  evidently  a  parabola  whose 
axis  is  the  axis  of  Y. 
Ex.  4.  Trace  the  curve  whose  equation  i&^\=:^^.3x. 

91.  To  find  the  equation  to  the  tangent  at  any  jpoint  of  a 
jpardbola. 

Let  the  equation  to  the  parabola  be  y'—^ax. 

Let  x'^  y'  be  the  co-ordinates  of  the  point  on  the  curve  at 
which  the  tangent  is  drawn,  and  x'\  y"  the  co-ordinates  of  an 
adjacent  point  on  the  curve.  The  equation  to  the  secant  line 
passing  through  the  points  x\  y'  and  x" ^  y"  (Art.  40)  is 

y-y'=^U^-^')-  (1) 

Now,  since  the  points  x' ^  y'  and  x" ^  y"  are  both  on  the  parabo- 
la, we  must  have  y'^=4:ax\ 
and                                       y"'^=^ax". 
Hence                          y"'-y''  =  ^a{x" -x'\ 

or  y"-y' ^  ^^ 

x"—x'    y"-[-y' 
Substituting  this  value  in  equation  (1),  the  equation  of  the 
secant  line  becomes 

y-y'=^^T:^'i«'-«')-  (2) 

The  secant  will  become  a  tangent  when  the  two  points  coin- 
cide, in  which  case  y' —y". 
Equation  (2)  will  then  become 

2« 

w^hich  is  the  equation  to  a  tangent  at  the  point  x\  y' . 
Clearing  of  fractions  and  transposing,  we  obtain 
yy' =  ^a{x-x')^■y'\ 


THE   PAEABOLA. 


89 


yy'  z='2ax— 2ax'  -\-  4:ax\ 
or  yy'  =  2a{x-^x% 

which  is  the  simplest  form  of  the  equation  to  the  tangent  line. 


92.  Points  where  the  tangent  cuts  the  axes 
the  point  in  which  the  tangent  intersects 
the  axis  of  X,  we  make  y= 0,  which  gives 

0  =  '2a{x-\-x')\ 
that  is,  x=—x\ 

or  AT=z-AR 

To  determine  the  point  in  which  the 
tangent  intersects  the  axis  of  Y,  we 
make  a?=0,  which  gives 

■    ■    y'-    y' 


To  determine 


"^ax' 


that  is, 


y 

AB 


93.  Definition.  A  suhtangent  to  a  parabola  is  that  part  of 
the  axis  intercepted  between  a  tangent  and  ordinate  drawn  to 
the  point  of  contact.  Thus  TK  is  the  subtangent  correspond- 
ing to  the  tangent  PT. 

From  Art.  92  we  see  that  the  suhtangent  to  the  axis  is  bisect- 
ed by  the  curve. 


94.  The  preceding  property  enables  us  to  draw  a  tangent  to 
the  curve  through  a  given  point.  Let  P  be  the  given  point ; 
from  P  draw  PR  perpendicular  to  the  axis,  and  make  AT= AR. 
Draw  a  line  through  P  and  T,  and  it  will  be  a  tangent  to  the 
parabola  at  P. 

95.  To  find  the  equation  to  a  tangent  to  the  jpardbola  in 
terms  of  the  tangent  of  the  angle  it  makes  with  the  axis. 

In  the  equation  of  a  tangent  line, 

'2a 
y-V=-zr{^-^')  (Art.  91,  Eq.  3), 

2a  .  y 

—J  represents  the  trigonometrical  tangent  of  the  angle  which 


90  ANALYTICAL   GEOMETEY. 

the  tangent  line  makes  with  the  axis  of  the  parabola  (Art.  38). 
If  we  represent  this  tangent  bj  m,  we  shall  have 

2<^  -.  2/'     ^  /.; 

— =m,  and  77=— .  (1) 

The  equation  to  a  tangent  line  to  the  parabola  (Art.  91)  is 

yy' :=^^a{x^x'\  •_        •      . 

,  ^a      2ax' 

whence  y=—,x-\- — r-, 

J       y'     ^    y'    ^ 

_2a      ^ax' 

2a      y' 

Hence,  substituting  equation  (1),  we  have 
-^  a 

which  is  the  equation  to  a  tangent  line. 
Hence  the^traight  line  whose  equation  is 


2/=mB+£,  Ar-^  CM^  ^W^I  -^ 


touches  the  parabola  whose  equation  is  y'^=^ax. 

Ex.  1.  Find  the  equation  of  a  tangent  to  the  parabola  2/*= 18a? 
at  the  point  x'  =  2,y'=zQ. 

Ex.  2.  Find  the  equation  of  a  tangent  to  the  parabola  y"^  =  4:Xy 
and  parallel  to  the  right  line  whose  equation  is  y=5x-{-l. 

Ex.  3.  On  a  parabola  whose  equation  is  y^  =  10xy  a  tangent 
line  is  drawn  through  the  point  whose  ordinate  is  8.  Deter- 
mine where  the  tangent  line  meets  the  two  axes  of  reference. 

Ex.  4.  On  a  parabola  whose  latus  rectum  is  10  inches,  a  tan- 
gent line  is  drawn  through  the  point  whose  ordinate  is  6  inch- 
es, the  origin  being  at  the  vertex  of  the  axis.  Determine  where 
the  tangent  line  meets  the  two  axes  of  reference. 

Ex.  5.  Find  the  angle  which  the  tangent  line  in  the  last  ex- 
ample makes  with  the  axis  of  X. 

Ex.  6.  On  a  parabola  whose  latus  rectum  is  10  inches,  find 
the  point  from  which  a  tangent  line  must  be  drawn  in  order 
that  it  may  make  an  angle  of  35°  w^ith  the  axis  of  the  parabola. 


THE   PARABOLA. 


91 


96.  Definitions.  The  term  normal  is  often  used  to  denote 
that  part  of  the  normal  line  (Art.  76)  which  is  included  be- 
tween-the  curve  and  the  axis  of  abscissas. 

A  subnormal  is  the  portion  of  the  axis  intercepted  between 
the  normal  and  the  ordinate  drawn  from  the  same  point  of  the 


curve. 


97.  To  find  the  equation  to  the  normal  at  any  jpoint  of  a   \ 
jparabola. 

Let  x\  y'  be  the  co-ordinates  of  the  given  point. 

The  equation  to  a  straight  line  passing  through  this  point 
(Art.  38)  is  y—y' =m{x — x') ; 

and,  since  this  line  must  be  perpendicular  to  the  tangent  whose 
equation  is 

2/-2/=^(^-^0(Art.91,Eq.3), 


y 


y 


^^=-§^(Art.45). 


we  have 

Hence  the  equation  to  the  normal  is 


2'-^--|(^-*')- 


98.  Point  where  the  normal  cuts  the  axis  ofx.  To  find  the 
point  in  which  the  normal  intersects  the  axis  of  abscissas,  make 
2/=0  in  the  equation  to  the  normal, 
and  we  have,  after  reduction, 

x—x'=z2a. 
But  X  is  equal  to  the  distance  AN, 
and  x'  to  AR;  hence  x—x'  is  equal 
to  EK,  w^hich  is  equal  to  2a;  that  is, 
the  subnormal  is  constant,  and  is 
equal  to  half  the  latus  rectum. 

Ex.  1.  On  a  parabola  whose  latus  rectum  is  10  inches,  a  nor- 
mal line  is  drawn  through  the  point  whose  ordinate  is  6  inches. 
Determine  where  the  normal  line,  if  produced,  meets  the  two 
axes  of  reference.  / ) 


^    kA^, 


92 


ANALYTICAL   GEOMETEY. 


Ex.  2.  Find  the  point  on  the  curve  of  a  parabola  whose  latus 
rectum  is  10  inches,  from  which,  if  a  tangent  be  drawn,  and  also 
an  ordinate  to  the  axis  of  X,  they  will  form  with  the  axis  a  tri- 
angle whose  area  is  36  inches. 

*^^{  99.  If  a  tangent  to  the  jparahola  cuts  the  axis  jproduced^the 
points  of  contact  and  intersection  are  equally  distant  from 
the  focus. 

Let  PT  be  a  tangent  to  the  parabola 
at  P,  and  let  PF  be  the  radius  vector 
drawn  to  the  point  of  contact. 
We  have  found  (Art.  92) 
TA=AR 
Hence  TF=AE  + AF  =  FP  (Art.  87) ; 
that  is,  the  distance  from  the  focus  to 
the  point  where  the  tangent  cuts  the 
axis,  is  equal  to  the  distance  from  the 


focus  to  the  point  where  the  tangent  touches  the  curve. 


100.-4  tangent  to  the  curve  makes  equal  angles  with  the  ra- 
dius vector  and  with  a  line  drawn  through  thejpoint  of  contact 
jparallel  to  the  axis. 

Let  TT^  touch  the  parabola  at  P,  and  let  BP  be  drawn 
thr(^ugh  P  parallel  to  AX ;  then  the  angle  BPT'  is  equal  to  the 
angle  ATP.  But  since  TF=PF,  the  angle  FTP  is  equal  to 
the  angle  FPT.  Hence  FPT  is  equal  to  BPT',  or  the  two  lines 
FP  and  BP  are  equally  inclined  to  the  tangent. 

101.  H  a  ray  of  light,  proceeding  in  the  direction  BP,  be  in- 
cident on  the  parabola  at  P,  it  will  be  reflected  to  F  on  account 
of  the  equal  angles  BPT^  and  FPT.  In  like  manner,  all  rays 
coming  in  a  direction  parallel  to  the  axis,  and  incident  on  the 
curve,  will  converge  to  F.  Also,  if  a  portion  of  the  curve 
revolves  round  its  axis  so  as  to  form  a  hollow  concave  mirror, 
all  rays  from  a  distant  luminous  point  in  the  direction  of  the 
axis  will  be  concentrated  in  F.     Thus,  if  a  parabolic  mirror  be 


THE   PARABOLA.  93 

held  with  its  axis  pointing  to  the  sun,  an  intense  heat  and  a 
brilhant  light  will  be  found  at  the  focus. 

102.  If  from  the  focus  of  a  jparahola  a  straight  line  he 
drawn  perpendicular  to  any  tangent^  it  will  intersect  this  tan- 
gent on  the  tangent  at  the\'ertex. 

Let  the  tangent  FT  be  drawn,  and  from 
the  focus  F  let  FB  be  drawn  perpendicu- 
lar to  it ;  the  point  B  will  fall  on  the  axis 
AY,  which  touches  the  curve  at  A  (Art. 

m). 

Since  the  triangle  FFT  is  isosceles,  the 
line  FB,  drawn  perpendicular  to  the  base 
FT,  w^ll  pass  through  its  middle  point ; 
and  since  AT  =  AR  (Art.  92),  the  line  AY,  which  is  parallel  to 
FR,  also  passes  through  the  middle  point  of  FT ;  that  is,  the 
line  FB  intersects  FT  in  the  same  point  with  AY. 

Since  the  triangle  FBT  is  right  angled  at  B,  we  have 
FB^^^FAxFTrrFAxFF, 
or  the  jperjpenidiculaT  from  the  focus  to  any  tangent  is  a  mean 
jprojportional  hetween  the  distances  of  the  focus  from  the  ver- 
tex and  the  point  of  contact. 

103.  To  determine  the  co-ordinates  of  the  points  of  inter- 
section of  a  straight  line  with  a  parabola. 

Let  the  equation  to  the  parabola  be 

f=^ax,  (1) 

and  the  equation  to  the  straight  line  be 

y=mx  +  c.  (2) 

As  in  Art.  78,  we  may  regard  (1)  and  (2)  as  simultaneous 
equations,  containing  but  two  unknown  quantities.  By  substi- 
tution in  equation  (1),  we  obtain 

Qny'^=.4:ay—^ac. 
Completing  the  square,  we  obtain 

y= — ±  —(a  —  amcf ; 


m     m^ 


t'l.'W 


94  a.nai;ytical  geometry. 

and,  since  y  lias  two  values,  we  conclude  that  there  will  be  two 
points  of  intersection. 

If  a—mc^  the  two  values  of  y  become  equal,  and  the  straight 
line  will  touch  the  parabola.  If  a—mc  is  negative,  the  straight 
line  will  not  meet  the  parabola. 

Ex.  1.  Find  the  co-ordinates  of  the  points  in  which  the  parab- 
ola whose  equation  is  y^—^x  is  intersected  by  the  line  whose 
equation  is  ?/=2a?— 5. 

Arts.  ?/=: 4.3166,  or  -2.3166;  i«=4.6583,  or  1.3417. 

Ex.  2.  Find  the  co-ordinates  of  the  points  in  which  the  parab- 
ola whose  equation  is  y^  —  X^x  is  intersected  by  the  line  whose 
equation  is  ?/=2a?— 5. 

Ans.  y^\%^Ta,  or  -3.5777;  aj= 8.7888,  or  0.7111. 

Ex.  3.  Find  whether  the  parabola  whose  equation  is  y^  —  l^x 
is  intersected  by  the  line  whose  equation  is  2/=i2aj+2,  and,  if 
there  is  a  point  of  contact,  determine  its  co-ordinates. 

Ex.  4.  Find  whether  the  parabola  whose  equation  is  y^  =  \^x 
is  intersected  by  the  line  whose  equation  is  2/=2a?+5. 

104.  To  determine  the  co-ordinates  of  the  joints  of  inter- 
section of  a  circle  and /parabola. 

If  the  centre  of  the  circle  is  not  restricted  in  position,  there 
may  be  four  points  of  intersection,  corresponding  to  an  equa- 
tion of  the  fourth  degree,  which  can  not  generally  be  resolved 
by  quadratics.  If,  however,  the  centre  of  the  circle  is  upon 
the  axis  of  the  parabola,  the  several  points  of  intersection  may 
be  easily  found. 

Let  the  equation  to  the  parabola  be 
y''  =  4cax, 
and  the  equation  to  the  circle  be 

x'-\-y'^r'', 
then,  by  substitution,  we  have 

x^-\-4:ax=r''^ 
and  £C==  -2a±(4»''+r')2, 

where  x  has  two  values,  but  one  of  them  is  negative,  and  gives 
imaginary  values  for  y.     There  will,  therefore,  be  but  two  real 


THE   PAEABOLA.-  95 

points  of  intersection.    These  have  the  same  abscissa,  and  their 
ordinates  will  differ  only  in  sign. 

Ex.  1.  Find  the  co-ordinates  of  the  points  in  which  the  parab- 
ola whose  equation  is  if  —  4:X  is  intersected  by  the  circle  whose 
equation  is  x'+f  =  U.  A7is.  £^=6.2462 ;  y=  d=  4.9985. 

Ex.  2.  Find  the  co-ordinates  of  the  points  in  which  the  parab- 
ola whose  equation  is  y^  =  18x  is  intersected  by  the  circle  whose 
equation  is  x^-^]f— 32£c— 40. 

x—^,  or  10  : 


Ans.  ,  ,    ,^  ^    ,^ 

?/=±6-v/2,  or  d=6/D. 

Construct  the  two  curves,  and  show  the  points  of  intersec- 
tion. 

Ex.  3.  Find  the  co-ordinates  of  the  points  in  which  the  parab- 
ola whose  equation  is  if  —  2x  is  intersected  by  the  circle  whose 
equation  is  x''+y^  =  6x-\-6. 

J^Cu  *  105,  To  transform  the  equation  to  tJie  parabola  into  anotk 
er  referred  to  oblique  axes^  and  so  that  the  equation  shall  pre- 
serve the  same  form. 

The  formulas  for  passing  from  rectangular  to  oblique  axes 
(Art.  56)  are  x—m-\-x'  cos.  a-\-y'  cos.  /3, 

y=n-\-x'  sin.  a-\-y'  sin.  /3. 

Substituting  these  values  in  the  equation  y'^—^ax^  and  ap 
ranging  the  terms,  we  have 

f  sin.^^-f  a?'^  sin.'''a  +  2a?y  sin.  a  sin.  j3-f 
■  2(^  sin.  (5  — 2a  cos.  (i)y'-\-n''—4:am=^2{2a  cos.  a— 7^  sin.  a)x', 
which  is  the  equation  to  the  parabola  referred  to  any  oblique 
axes. 

In  order  that  this  equation  may  be  of  the  form  y^=4:ax,  we 
must  have  the  following  conditions  : 

1st.  There  must  be  no  absolute  term  ;  hence  n^—4:am=0. 

2d.  There  must  be  no  term  containing  x'^;  hence  sin.  'a=0. 

3d.  There  must  be  no  term  containing  ccy  ,•  hence  sin.  a  sin. 

j3=a 

4th.  There  must  be  no  term  containing  y^;  hence  n  sin.  /3— 
2ojcos.j3=0. 


96 


ANALYTICAL   GEOMETRY. 


These  equations  contain  four  arbitrary  constants,  m,  n,  a,  ]3  / 
it  is  therefore  possible  to  assign  such  values  to  the  constants 
as  to  satisfy  the  four  equations,  and  thus  reduce  the  new  equa- 
tion of  the  parabola  to  the  proposed  form. 

Since  the  equation  'i/=4:ax  becomes  n'=4:am  by  substituting 
the  co-ordinates  of  the  new  origin  for  x  and  y,  it  follows  that 
the  first  condition,  ^'— 4am=0,  requires  the  new  origin  to  he 
on  the  curve. 

The  second  condition,  sin.''a=0,  requires  the  new  axis  ofx 
to  he  parallel  to  the  axis  of  the  jparahola. 

The  third  condition,  sin.  a  sin.  j3  =  0,  is  satisfied  by  the  sec- 
ond, without  introducing  any  new  condition. 

2^       2<z 
Since  --  or  —  (Art.  95)  has  been  found  to  represent  the  tan- 

if 

gent  of  the  angle  which  the  tangent  line  makes  with  the  axis 
of  the  parabola,  the  fourth  condition,  n  sin.  (5— 2a  cos.  /3  =  0,  or 

■^    '|:— tang.  j3= — ,  requires  that  the  new  axis  ofj  shall  he 

tangent  to  the  curve  at  the  origin. 

Y'      ^.^cr^'  If  J  therefore,  the  curve  is 

referred  to  any  tangent  line 
A' Y',  and  a  line  A'X^  drawn 
^'  through  the  point  of  contact 
parallel  to  the  axis,  the  equa- 
tion becomes 

2{2a  COS.  a—n  sin.  a)x^) 
or,  since  sin.  a=0, 
and  cos.  a=l, 


we  have 


y  = 


Aa 


sin.=/3 


If  w^e  represent  -: — ^  bv  4:a\  and  omit  the  accents  of  the  va- 
riables, we  shall  have        *       ^ 

y'=4:a'x, 
which  is  the  equation  required. 


THE   PAKABOLA.  97 

106.  Since  the  preceding  problem  furnished  four  arbitrary 
constants,  m,  n,  a,  j3,  and  required  but  three  independent  con- 
ditions (the  second  and  third  being  but  one),  we  may  assign 
any  value  at  pleasure  to  either  of  them  except  a ;  that  is,  the 
new  origin  may  hejplaced  any  where  on  the  curve. 

107.  From  the  equation  of  Art.  105,  we  find 

y=^^/^.a'x^ 
which  shows  that  for  every  positive  value  of  x  there  are  two 
values  of  y  equal  numerically  but  having  opposite  signs,  and 
these  two  values,  taken  together,  form  a  chord  PP^  parallel  to 
the  axis  of  Y,  which  chord  is  bisected  by  the  axis  of  X  at  M. 
So,  also,  the  parallel  chord  QQ'  is  bisected  by  the  axis  of  X  at 
E".  Hence' a  straight  line  parallel  to  the  axis  of  the  jparabola 
hisects  all  chords  parallel  to  the  tangent  at  its  extremity. 

108.  Definition,  A  diameter  of  a  parabola  is  a  straight  line 
drawn  through  any  point  of  the  curve  parallel  to  the  axis  of 
the  parabola.  The  vertex  of  the  diameter  is  the  point  in  which 
it  meets  the  curve. 

109.  The  equation  of  Art.  105,  y'^  =  Mx,  is  called  the  equa- 
tion of  the  parabola  referred  to  a  tangent  line,  and  the  diame- 
ter drawn  through  the  point  of  contact ;  and,  since  the  new 
axis  of  Y  is  a  tangent  to  the  curve  at  the  origin,  a  diameter  hi- 
sects all  chords  parallel  to  the  tangent  at  its  extremity. 

The  equation  y^z=z^a'x 

ehows  that  for  oblique  axes  the  squares  of  the  ordinates  are 
proportional  to  the  corresponding  abscissas^  which  is  a  gener- 
alization of  the  property  proved  in  Art.  89. 

110.  To  determine  the  value  of  the  coefficient  of  is.  in  the 
^uation  y" = ^a'x. 

From  Art.  105  we  have 

sin.  /3_2« 

COS.  /3 ""  71  ' 

E 


98  ANALYTICAL    GEOMETRY. 

whence  n  sin.  j3=2a  cos.  /3, 

and  n^  sin.  "jS^^^^"  cos.  'j3, 

^4t^'_4a''sin.'/3. 

Therefore  sin.  '/3 = ^._^^^a. 

But  from  the  equation  to  the  curve  referred  to  the  original 
axes  we  have  v^=^am; 

therefore  sm.  ^=4^^+4^»=^^, 

and  -. — ^7^=a+7n=a , 

sm.  ^/3 

But  m  represents  the  abscissa  of  the 

new  origin  referred  to  the  original  axes ; 

hence 

«+m=FA'  (Art.  87)  =  a', 

or  the  coefficie^nt  of  x  in  the  equation 

y^=4:a^x  \&fouT~times  the  distance  from 

the  focus  to  the  new  origin. 

111.  To  determine  the  length  of  the  chord  drawn  through 
the  focus  jparallel  to  the  new  axis  of  ordinates. 

If  through  the  focus  F  the  line  BD  be  drawn  parallel  to  the 
new  axis  of  Y,  then,  calling  x  and  y  the  co-ordinates  of  the 
point  D,  we  have 

i»=A'C=TF= AT  (Art.  99)=«'  (Art.  110). 
But,  by  Art.  105,  y' = 4ca'x  ; 

hence  y" = ^a'  xa'= ^a'*, 

or  y='^a\ 

and  ^y—^a'\ 

that  is,  the  coefficient  ^a'  is  the  double  ordinate  passing  through 
the  focus  and  corresponding  to  the  diameter  which  passes 
through  the  origin, 

112.  Definition.  The  parameter  of  any  diameter  is  the 
double  ordinate  which  passes  through  the  focus. 


THE   PARABOLA. 


99 


From  Art.  110  we  see  that  the  parameter  of  any  diamete? 
is  equal  to  four  times  the  distance  from,  the  vertex  of  that  di- 
ammeter  to  the  focus. 

Ill  the  equation  y^^^a'x^  ^a'  is  the  parameter  of  the  diame- 
ter passing  through  the  origin. 

The  parameter  to  the  axis  is  called  the  principal  parameter, 
or  latus  rectum  (Art.  84). 

113.  To  find  the  polar  equation  to  the  parabola^  the  focus 
heing  the  pole. 

Let  the  axis  be  the  initial  line.     Repre- 
sent FP  bj  p,  and  PFN  by  Q. 
Then  PF=:PD  =  BF  +  FI^, 

or  |0==2^+^  cos.  0\ 

whence  /o(l  — cos.''0)=2<aj, 

2a 

which  is  the  polar  equation  to  the  parabola. 


114.  If  0=180°,  then  cos.  d- 
comes 

2a 


=  — 1,  and  the  value  of  p  be- 
^=FA. 


If  0=90°,  then  cos.  0=0,  and  the  value  of  p  becomes 

p=2a=.YQ. 
If  0=0,  then  cos.  0=1,  and  we  have 

2a 

the  radius  vector  takes  the  direction  AX,  and  does  not  meet 
the  curve  at  a  finite  distance. 


115.  If  the  variable  angle  be  measured  from  the  point  A  to- 
ward the  right,  then  we  must  substitute  for  0, 180°— 0,  in  which 
case  COS.  0=— cos.  0',  and  w^e  have 

2a 
^~H-cos.0'- 


100 


ANALYTICAL   GEOMETRY. 


Ex.  1.  What  is  the  polar  equation  of  a  parabola  whose  latus 
rectum  is  10,  the  pole  being  at  the  focus ;  and  what  is  the 
length  of  the  radius  vector  for  0=60°  ? 

Ex.  2.  The  latus  rectum  of  a  parabola  is  8  inches,  and  9= 
135° ;  what  is  the  radius  vector  ? 

Ex.  3.  The  latus  rectum  of  a  parabola  is  6  inches,  and  the 
radius  vector  is  10  inches  ;  determine  the  value  of  0. 

Ex.  4.  The  radius  vector  of  a  parabola  is  25  inches  and  0= 
135° ;  what  is  the  latus  rectum  ? 


116.  To  determine  the  area  of  a  segment  included  hetween 
an  arc  of  a  parabola  and  a  cJiord  jperjpendicular  to  the  axis, 
y^ ^--^^      1^6^  PAQ  be  a  segment  of  a  parab- 
ola, bounded  by  the  curve  PAQ,  and 
the  chord  PQ  perpendicular  to  the 
axis  AE.    It  is  required  to  determine 

its  area. 

Inscribe    in    the    semi  -  parabola 
PAR  a  polygon  PPT''. . . .  AR,  and 


f 


Xy" 


through  the  points  P,  P',  V" ^  etc., 

draw  parallels  to  AP  and  PR,  form- 

N  ^^^  ing  the  interior  rectangles  P'R,P'T', 

etc.,  and  the  corresponding  exterior  rectangles  P'M,  P'^M',  etc. 

Designate  the  former  by  P,  P',  P'',  etc. ;  the  latter  hyp,p', 

j>'\  etc.,  and  the  corresponding  co-ordinates  by  x,  y,  x\  y\  etc. 

We  shall  then  have 

PTv^P'R'xRR^ 

V=y\x-x'). 

P'M=:P'M'x]\OP, 

J[>=x'{y-y'). 

V_y\x-x') 


the  rectangle 

or 

Also  the  rectangle 

or 

Whence 


■x\y-y')'  (^) 

But,  since  the  points  P,  P',  etc.,  are  on  the  curve,  we  have 
y''=^ax,  y'''—^ax'\ 


whence 


)^x'=~r^,  and  aj'=|-. 
^a    '  4a 


THE   PARABOLA.  101 

Substituting  these  values  in  equation  (1),  we  obtain 

^j/{y'-y'l_y+y' _^    y 
jp    y"\y-y)     y'         y'' 

In  the  same  manner  we  find 

P'   .    y' 

1  4-  — 

!>'"     y"' 
P''   .    y'' 

y7  =  l+^/jetc. 

If  now  we  suppose  the  vertices  of  the  polygons  P,  P',  P^', 
etc.,  to  be  so  placed  that  the  ordinates  shall  be  in  geometrical 
progression,  we  shall  have 

y.-t-yl  etc 

SO  that  each  interior  rectangle  has  to  its  corresponding  exterior 

y 

rectangle  the  ratio  of  1+—  to  1. 

if 

Therefore,  by  composition, 

P-l-F^  +  F^  +  ^etc.,_     ,  y 
i>+y +//'+,  etc.,        '^y" 
that  is,  the  sum  of  all  the  interior  rectangles  is  to  the  sum  of 

y 

all  the  exterior  rectangles  as  1  +  —  to  1. 

When  the  points  P,  P',  V\  etc.,  are  taken  indefinitely  near, 

y 

the  ratio  —,  approaches  indefinitely  near  to  a  ratio  of  equality ; 

if 

the  sum  of  the  interior  rectangles  converges  to  the  area  of  the 
interior  parabolic  segment  APP,  and  the  sum  of  the  exterior 
rectangles  to  the  area  of  the  exterior  parabolic  segment  AMP. 
Designating  the  former  by  S,  and  the  latter  by  5,  we  have 

or  S  =  25=|(S  +  s). 

But  S+«  is  equal  to  the  area  of  the  rectangle  AMPP;  hence 
the  parabolic  segment  APP  is  two  thirds  of  the  rectangle 
AMPP,  or  the  segment  PAQ  is  two  thirds  of  the  rectangle 
PMNQ.    Hence  the  area  of  ajparaboliG  segment  cut  off  by  a 


102  ANALYTICAL   GEOMETRY. 

double  ordinate  to  the  axis  is  two  thirds  of  the  circumscribing 
rectangle. 

Ex.  1.  Determine  the  area  of  the  parabolic  segment  cut  off 
by  a  double  ordinate  whose  length  is  24:  inches,  the  latus  rect- 
um being  8  inches. 

Ex.  2.  The  area  of  a  parabolic  segment  cut  off  by  a  double 
ordinate  to  the  axis  is  96,  and  the  corresponding  abscissa  is  6. 
Determine  the  equation  to  the  curve. 

117.  By  a  demonstration  like  that  of  the  preceding  article, 
it  may  be  also  shown  that  the  area  of  a  parabolic  segment  cut 
off  by  the  doublie  ordinate  of  aiiy  diameter  is  two  thirds  of  the 
circumscribing  parallelogram . 

Example.  Prove  that  if  two  tangents  are  drawn  at  the  ex- 
tremities of  any  chord  of  a  parabola,  the  segment  cut  off  from 
the  parabola  is  two  thirds  of  the  triangle  formed  by  the  choi-d 
and  the  two  tangents. 


Ox/]Sl\   m^'Mn 


y 


THE   ELLIPSE.  103 


SECTION^  YL 

THE   ELLIPSE. 

118.  An  ellipse  is  a  plane  curve  traced  out  by  a  point  which 
moves  in  such  a  manner  that  the  sum  of  its  distances  from  two 
fixed  points  is  always  the  same.  The  two  fixed  points  are  call- 
ed iliQfoci  of  the  ellipse. 

Thus,  if  F  and  F'  are  two  fixed  points, 
and  if  the  point  P  moves  about  F  in 
such  a  manner  that  the  sum  of  its  dis- 
tances from  F  and  F^  is  always  the  same, 
the  point  P  will  describe  an  ellipse,  of 
which  F  and  F'  are  the  foci.  The  dis- 
tance of  the  point  P  from  either  focus 
is  called  \hQ  focal  distance^  or  the  radius  ve<itor, 

119.  Description  of  the  curve.  From  the  definition  of  an 
ellipse  the  curve  may  be  described  mechanically.  Thus,  take 
a  thread  whose  length  is  greater  than  the  distance  FF',  and 
fasten  one  of  its  extremities  at  F,  the  other  at  Fl  Place  the 
point  of  a  pencil,  P,  against  the  thread,  and  slide  it  so  as  to 
keep  the  thread  constantly  stretched ;  the  point  of  the  pencil 
will  describe  an  ellipse.  For  in  every  position  of  P  we  shall 
have  FP+FT  equal  to  the  fixed  length  of  the  thread;  that  is, 
equal  to  a  constant  quantity. 

120.  Definitions,  The  centre  of  the  ellipse  is  the  middle 
point  of  the  straight  line  joining  the  foci. 

A  diameter  is  any  straight  line  passing  through  the  centre, 
and  terminated  on  both  sides  by  the  curve. 

Tlie  diameter  which  passes  through  the  foci  is  called  the 
transverse  axis,  or  the  major  axis. 

The  diameter  which  is  perpendicular  to  the  major  axis  is 
called  the  conjugate  axis,  or  the  minor  axis. 


104  ANALYTICAL   GEOAEETKY. 

The  latus  rectum  is  the  chord  drawn  through  one  of  the  foci 
perpendicular  to  the  major  axis. 

121.  To  find  the  equation  to  the  ellipse  referred  to  its  axes. 

Y  Let  r  and  F^  be  the  foci,  and 

draw  the  rectangular  axes  CX, 

.7^*"\  p  CY,  the  origiii,  C,  being  placed 

^^^^^"-^^Z^  at  the  middle  of  FF;    Let  P  be 

FR/A — ^  ^^y  point  of  the  curve,  and  draw 

y  PR  perpendicular  to  CX. 

_,,-^  Let  2c  denote  FF',  the  con- 

^'  stant  distance  between  the  foci, 

and  2a  denote  FP+F'P,  the  constant  sum  of  the  focal  distan- 
ces.    Denote  FP  by  r,  FT  by  r\  and  let  x  and  y  denote  the 
co-ordinates  of  the  point  P. 
Then,  since 

FP=PR'^-fEF'=PR='4-(CE-CF)', 
we  have  r''-=y'-\-{x— c)\  (1) 

Also,      PF^" = PE=^ + PF'^ = PR^  -f  (CR  -f  CF)^ 
That  is,  r''=y''-\-{x^-c)\  (2) 

^    Adding  equations  (1)  and  (2),  we  obtain 
ts^  r'  +  r'^^^iy'+x'^c')',  (3) 

i    ^and  subtracting  equation  ('1)  from  (2),  we  obtain 

r'^—r'^=4:cx, 
which  may  be  put  under  the  form 

{r'  -h  7'){r' —r)= 4:cx.  (4) 

But  from  the  definition  of  the  ellipse  we  have 
^;;  r'+r=2a.  (5) 

sV^.  Dividing  equation  (4)  by  equation  (^,  we  obtain 
X\  ^  2cx 

a 
Combining  the  last  two  equations,  we  find 

r'=a+'-^,  (6) 


THE   ELLIPSE.  105 

Squaring  these  values,  and  substituting  them  in  equation  (3), 
we  obtain 

which  may  be  reduced  to  the  form  y^ 

a\f-\-{a'-c')x'=a\a'-c'),  (8) 

which  is  the  equation  to  the  ellipse. 

This  equation  may,  however,  be  put  under  a  more  convenient 
form. 

Represent  the  line  BC  by  h.     In  the  two  right-angled  trian- 
gles BCF,  BCF',  CF  is  equal  to  CF',  and  BC  is  common  to 
both  triangles ;  hence  BF  is  equal  to  BF^.     But,  by  the  defini- 
tion of  the  ellipse,  l^Y -{•'SF'  —  2a ;  consequently  BF=«. 
Now  BC'=BF^-CF^; 

that  is,  h''=a'-c\  (9) 

Substituting  this  value  in  equation  (8),  we  obtain 

a\/^Jfx'=a'h%  (10) 

x^    it 
or  ^+1^=1'  W 

which  is  the  equation  of  the  ellipse  referred  to  its  axes. 

This  equation  is  sometimes  written 

2/'=|(«'_a;').  (12) 


122.  Points  of  intersection  with  the  axes.  To  determine 
where  the  curve  cuts  the  axis  of  X,  make  ?/=0  in  the  equation 
of  tlie  ellipse,  and  we  obtain 

aj=:±a=:CAor  CA', 
which  shows  that  the  curve  cuts  the  axis  of  abscissas  in  two 
points,  A  and  A',  at  the  same  distance  from  the  origin,  the  one 
being  on  the  right,  and  the  other  on  the  left ;  and,  since  2CA, 
or  AA',  is  equal  to  2«,  it  follows  that  the  sum  of  the  two  lines 
drawn  froin  any  jpoint  of  an  ellijpse  to  the  foci  is  equal  to  the 
major  axis. 

If  we  make  aj=0  in  the  equation  of  the  ellipse,  we  obtain 

y=±J,  =CBorCB', 

E2 


106 


ANALYTICAL   GEOMETEY. 


which  shows  that  the  curve  cuts  the  axis  of  Y  in  two  points, 
B  and  B',  at  the  same  distance  from  the  origin. 

123.  Curve  traced  through  intermediate  points.  If  we  wish 
to  trace  the  curve  through  the  intermediate  points,  we  reduce 
the  equation  to  the  form 


y=^-Va'-x\ 


a 


from  which  we  may  compute  the  vahie  of  y  corresponding  to 
any  assumed  value  of  x. 

Exam/pie.  Trace  the  curve  whose  equation  is 
492/'^  +  36aj'^=:lY64. 

Solving  the  equation  for  y,  we  have 


2/=  ±^1/49-0?'. 

By  assuming  for  x  different  values  from  0  to  7,  we  obtain 
the  corresponding  values  of  y  as  given  below. 
When  a;=0, 2/=  ±6.  When  aj=:4,  y^  ±4.92. 

aj=l,  y=±6.94.  ic=:5, 2/=±4.20. 

a?=r2, 2/=±5.75.  £ez=6,  ?/:=±3.09. 

a?=::3,  '?/=±5.42.  a;=r7, 2/=±0. 

When  x—O^y  will  equal  ±6, 
which  gives  two  points,  a  and  a', 
one  above  and  the  other  below 
the  axis  of  X.  When  x=l,y= 
±  5.94,  which  gives  the  points  b 
and  h'.  When  i«=2,  ^=±5.75, 
which  gives  the  points  c  and  c\ 
etc.  If  we  suppose  x  greater  than  7,  the  value  of  2/  will  be  im- 
aginary, which  shows  that  the  curve  does  not  extend  from  the 
centre  beyond  the  value  a? =7. 

If  a?  is  negative,  we  shall  in  like  manner  obtain  points  in  the 
third  and  fourth  quadrants,  and  the  curve  will  not  extend  to 
the  left  beyond  the  value  aj=  —  7. 

The  ellipse  is  seen  to  be  symmetrical  above  and  below  the 
axis  of  £C,  and  also  to  the  right  and  left  of  the  axis  of  y. 


THE   ELLIPSE.  107 

124.  The  circle  is  a  jparticulat  case  of  the  ellipse.    When  b 
is  made  equal  to  a,  the  equation  of  the  ellipse  becomes 

which  is  the  equation  of  a  circle  ;  hence  the  circle  may  he  re- 
garded as  an  ellipse  whose  two  axes  are  equal  to  each  other. 

125.  To  find  the  foci  of  an  ellipse  when  the  two  axes  are 

given.     Since  BF  or  BF'  is  equal  to  ^ Jb 

a  (Art.  121),  it  follows  that  the  dis- 
tance from  either  focus  to  the  extrem- 
ity of  the  minor  axis  is  equal  to  half  '^^ 
the  major  axis. 

If,  then,  from  B,  the  extremity  of 
tlie  minor  axis,  with  a  radius  equal  to  half  the  major  axis,  we 
describe  an  arc  cutting  the  major  axis  AA'  in  F  and  F',  the  two 
points  of  intersection  will  be  the  foci  of  the  ellipse. 

126.  To  find  the  length  of  the  latus  rectum.    According  to 
Art.  121,  Eq.  12, 

f~{a^-x^). 
Suppose  a? =c,  or  CF;  then 

where  y  is  the  ordinate  at  the  point  F.    But  by  Art.  121, 
Eq.9, 

hence  we  have 

or  a\h::h'.y,  ' 

and  2a:2h::2h:  2y. 

But  2y  here  represents  the  double  ordinate  drawn  through 
the  focus,  and  is  called  the  latus  rectum  (Art.  120) ;  hence  the 
latus  rectum  of  any  ellipse  is  a  third  vroportional  to  the  ma- 
jor and  minor  axes.      ^    .  ^-^ 


108  ANALYTICAL   GEOMETRY. 

127.  Equation  of  the  ellipse  in  terms  of  the  eccentricity, 

/I 
The  fraction  — ,  which  represents  the  ratio  of  CF  to  CA,  or  the 

distance  from  the  centre  to  either  focus,  divided  by  half  the 
major  axis,  is  called  the  eccentricity  of  the  ellipse.  If  we  rep- 
resent the  eccentricity  by  e,  then 

c 

-—e^  or  c=ae. 

But  we  have  seen  that  c^—d^—h^\ 
hence  d^—W  —  c^^^ 

or  ^  =  1—6. 

Making  this  substitution,  the  equation  of  the  ellipse  becomes 

which  is  the  equation  in  terms  of  the  eccentricity. 

128.  To  find  the  distance  of  any  jpoint  on  the  curve  from 
either  focus.    Equations  (6)  and  (Y)  of  Art.  121  are 

ex 

cx 

r=.a-^—. 

a 

ft 

Substituting  e  for  -  these  equations  become 

T'=a-\-ex^ 
T—a—ex, 
which  equations  represent  the  distance  of  any  point  on  an  el- 
lipse from  either  focus.  ^^ 
Multiplying  these  values  together,  we  obtain 

which  is  the  value  of  the  product  of  the  focal  distances. 

The  equation  of  an  ellipse  may  assume  forms  differing  from 
those  of  Art.  121,  in  consequence  of  multiplication  or  division 
by  a  constant,  or  of  transposition.  Thus,  ic'-f  ^''zrT;  2/''  =  25 
— Sa;";  4(a3'+2/')=:7+aj',  are  equations  of  ellipses  referred  to 
the  centre  and  axes. 


THE  ELLIPSE.  109 

Ex.  1.  Trace  the  curve  wliose  equation  is  Sx^-{'6y^=16. 

Ex.  2.  In  a  given  ellipse,  half  the  sum  of  the  focal  distances 
is  4,  and  half  the  distance  between  the  foci  is  3 ;  what  is  the 
equation  to  the  ellipse  ? 

Ex.  3.  In  a  given  ellipse,  the  sum  of  the  focal  distances  is  10, 
and  the  difference  between  the  squares  of  half  that  sum  and 
half  the  distance  between  the  foci  is  16 ;  what  is  the  equation 
to  the  ellipse  ? 

Ex.  4.  What  is  the  eccentricity  of  the  ellipse  whose  equation 

Ex.  5.  Trace  the  curve  whose  equation  is  x*-}- 4:7/ =16, 
Ex.  6.  Trace  the  curve  whose  equation  is  Sx^-\-4:y^=120. 
Ex.  7.  What  are  the  eccentricities  of  the  ellipses  of  examples 
6  and  6? 

129.  To  find  the  equation  of  the  ellijpse  when  the  origin  is 
at  the  vertex  of  the  major  axis.    The 
equation  of  the  ellipse  when  the  ori- 
gin is  at  the  centre  is 

f~{a-x^  (1) 

If  the  origin  is  placed  at  h! ,  the  or- 

dinates  will  have  the  same  value  as 

when  the  origin  was  at  the  centre,  but  the  abscissas  will  be 

changed.     If  w^e  represent  the  abscissas  reckoned  from  A'  by 

x\  we  shall  have  CR=:  A'K- A'C, 

or  x=x'—a. 

Substituting  this  value  of  x  in  equation  (1),  we  have 

which  is  the  equation  of  the  ellipse  referred  to  the  vertex  A'. 

130.  Relation  of  ordinates  to  the  major  axis.  If  the  last 
equation  be  resolved  into  a  proportion,  we  shall  have 

y^\{^a—x)x'.  '.b'^'.a^. 
Now  2a  represents  the  major  axis  A  A' ;  and  since  x  repre- 


110 


ANALYTICAL   GEOMETRY. 


sents  A^R,  'ia—x  will  represent  AR ;  therefore  {^a—x)x  repre- 
sents the  product  of  the  segments  into  which  the  major  axis  is 
divided  by  the  ordinate  PR.  Hence  we  have,  the  square  of 
any  ordinate  to  the  major  axis  of  an  ellipse,  is  to  the  product 
of  the  segTnents  into  which  it  divides  that  axis,  as  the  square 
of  the  minor  axis,  is  to  the  squai'e  of  the  major  axis. 

If  we  draw  a  second  ordinate  P'R'  to  the  major  axis,  we 
shall  have  PR^  A'R  x  RA): :  If :  a'^ :  P'R'^ :  k'R'  x  RA) 
or  PR^ :  P'R" : :  A'R  x  RA :  A'R'  x  fe'A ; 

that  is,  the  squares  ofordinates  to  the  major  axis  of  an  ellipse 
are  to  each  other  as  the  products  of  the  segments  into  which 
they  divide  that  axis. 


131.  Ordinates  to  the  minor  axis.  The  equation  to  the  el- 
lipse, Art.  121,  Eq.  10,  may  be  put  under  the 
form 

or  a':¥::x':{h-\-y)(h-y). 

Now  y  represents  CR;  hence  b-i-y  repre- 
sents B'R,  and  b—y  represents  BR.  Also  x 
represents  PR,  which  may  be  called  an  or- 
dinate'to  the  minor  axis.  Hence  we  have  the  square  of  any 
ordinate  to  the  minor  axis  of  an  ellipse,  is  to  the  product  of 
the  segments  into  which  it  divides  that  axis,  as  the  square  of 
the  major  axis,  is  to  the  square  of  the  minor  axis. 

Example.  The  major  axis  of  an  ellipse  is  12  inches,  and  the 


curve  passes  through  the  two  points  x 

y 


4, 2/=0,  and  x=—4:, 


0 ;  required  the  equation  of  the  ellipse. 


132.  An  ordinate  to  the  major  axis  of  an  ellipse  is  to  the 
corresponding  ordinate  of  the  circumscribed  circle,  as  the  mi- 
nor axis  is  to  the  major  axis. 

Let  a  circle  be  described  on  AA^  as  a  diameter,  and  let  the 
ordinate  PR  of  the  ellipse  be  produced  to  meet  the  circumfer- 
ence of  the  circle  in  V\ 


THE   ELLIPSE. 


Ill 


The  equation  of  the  ellipse  when 
the  origin  is  at  the  centre  (Art.  121, 
Eq.  12)  is 

But  {a—x){a-\-x)  represents  AKx 
A'R ;  hence  we  have 

PR''         h^ 
ARxA'R"^'" 
But  P'R''= AE  X  A^R  (Goom.,  B.  lY.,  Pr.  23,  Cor.) ;  hence 

PR''     h^ 


or 


PR:P^R::5:a::2J:2^. 


133.  An  ordinate  to  the  minor  axis  of  an  elli-pse  is  to  the 
corresponding  ordinate  of  the  inscribed  circle^  as  the  major 
axis  is  to  the  minor  axis. 

Let  a  circle  be  described  on  BB'  as  a  di- 
ameter, and  let  the  ordinate  PR  of  the  el- 
lipse meet  the  circumference  of  the  circle 
in  P; 

The  equation  of  the  ellipse  when  the  ori-  ^  I 
gin  is  at  the  centre  is 

«''=J(*'-2/')=J(J-2/)(*+y)- 

But  (J>—y)(J>+y)  represents  BRxB^R;  hence  we  have 

PR'         a" 
BRxB^R-r 
ButBRxB'R=P^R^;  hence 

PR'     a" 


or 


P^R'~5'' 
PR:P'R::a:J::2^:2i. 


C^<«f/  ^134.  To  find  the  equation  to  the  tangent  at  any  j^oint  of  an 
'  ellipse. 

Let  the  equation  to  the  ellipse  be  a^y^-^h^x^=a^h'^. 


112 


ANALYTICAL   GEOMETRY. 


Let  x\  y'  be  the  co-ordinates  of  the  point  on  the  curve  at 
which  the  tangent  is  drawn,  and  a?'',  y''  the  co-ordinates  of  an 
adjacent  point  on  the  curve.  The  equation  to  the  secant  line 
passing  through  the  points  x' ^  y'  and  x'^,  y"  (Art.  40)  is 


V   —V 


(1) 


Now,  since  the  points  x'^  y'  and  x" ^  y"  are  both  on  the  ellipse, 
we  must  have  aJ'y'^-\-h''x'^=a^¥, 

and  aY"-^h''x"'=a'h''', 

therefore,  by  subtraction,  a'((y"'-y")^h\x'"'~-x'^)  =  0, 

y"-y'        h^  x"+x' 
^^  x"-x'-~a''y"-^y" 

Substituting  this  value  in  equation  (1),  the  equation  of  the 
secant  line  becomes 

The  secant  will  become  a  tangent  when  the  two  points  coin- 
cide, in  which  case 

x'—x'\2,\\^y'=y'\ 
Equation  (2)  will  then  become 

Vx' 

which  is  the  equation  to  a  tangent  at  the  point  a?',  y'. 

Clearing  this  equation  of  fractions  and  transposing,  we  ob- 
tain a^yy'  +  I'xx' = ay''\'  h'x'' ; 
hence                    a'yy' +h^xx[=a'h%.       ..  ex^vi  JCC^kv  m       (4) 
which  is  the  simplest  form  of  the  equation  to  the  tangent  line. 


135.  Points  where  the  tangent  cuts  the  axes.     In  equation 

(4)  of  the  last  article,  x  and  y  are 
co-ordinates  of  any  point  of  the  tan- 
gent line.     Make  y=0,  in  which 
case  aj=CT,  and  we  have 
b'xx'=a'h'; 


that  is, 


x=- 


X' 


THE   ELLIPSE.  113 

But  x'  is  CK ;  hence     CR .  CT = C  A\ 

If  from  CT  we  subtract  CR  or  x\  we  shall  have  the  subtangent 

Since  the  subtangent  is  independent  of  the  minor  axis,  it  is  the 
same  for  all  ellipses  which  have  the  same  major  axis ;  and 
since  the  circle  on  the  major  axis  may  be  considered  as  one  of 
these  ellipses,  the  subtangent  is  the  same  for  an  ellipse  and  its 
circumscribing  circle. 

To  determine  the  point  in  which  the  tangent  intersects  the 
axis  of  y,  we  make  aj=0,  which  gives 

Therefore  CN".CT'=CB\ 

136.  To  draw  a  tangent  to  an  ellijpse  through  a  given  jpoint. 

Let  P  be  the  given  point  on 
the  ellipse.  On  AA'  describe 
a  circle,  and  through  P  draw 
the  ordinate  PR,  and  produce 
it  to  meet  the  circumference  of 
the  circle  in  P'.  Through  P' 
draw  the  tangent  PT,  and  from 
T,  where  the  tangent  to  the  cir- 
cle meets  the  major  axis  pro- 
duced, draw  PT ;  it  will  be  a  tangent  to  the  ellipse  at  P  (Art. 
135). 

^aA137.  To  find  the  equation  of  a  tangent  line  to  the  ellipse  in 
terms  of  the  tangent  of  the  angle  it  maJces  with  the  major  axis. 
In  the  equation  of  the  tangent  line  (Art.  134,  Eq.  3), 

V'x' 
—-5-7  represents  the  trigonometrical  tangent  of  the  angle 

if 

which  the  tangent  line  makes  with  tlie  major  axis  of  the  el- 


E^■ 


114  ANALYTICAL   GEOMETKY. 

lipse  (Art.  40).    If  we  represent  this  tangent  by  m,  we  shall 
have 

ay 

The  equation  of  the  tangent  line  (Art.  134,  Eq.  4)  was  re- 
duced to  the  form 

Vxx'    l^ 
Hence  11=— ^-f\ — 7, 

or  y=mxAr-,^ 

We  wish  then  to  express  —  in  terms  of  m, 

Now  Vx'  —  —  a^m, 

and  ay'^l^x'^^dV', 

4/3    3 

Therefore  aY'-\-^^^^^=a'b\ 

Hence  y'\a'm' +lf)=b\ 

and  -7zi:±-v/^WH-J\ 

y 

Hence  the  equation  to  the  tangent  may  be  written 

y — mx  ±  -y/a^w^ + V", 
Hence  the  straight  line  whose  equation  is 
y = mx  ±  V^^^m^  4-  b"", 
touches  the  ellipse  whose  equation  is  ay  +  b^x^=a'b*. 

Since  m  in  this  equation  is  indeterminate,  it  may  assume 
successively  any  number  of  values.  The  corresponding  straight 
lines  will  be  a  series  of  tangents  to  the  ellipse.  The  double 
sign  of  the  radical  shows,  moreover,  that  for  any  value  of  m 
there  are  two  tangents  to  the  ellipse  parallel  to  each  other. 

Ex.  1.  In  an  ellipse  whose  major  axis  is  50  inches,  the  ab-pp 
scissa  of  a  certain  point  is  15  inches,  and  the  ordinate  16  inch- x"^' 
es,  the  origin  being  at  the  centre.  Determine  where  the  tan-  7' 
gent  passing  through  this  point  meets  the  two  axes  produced.  ^^ 

Ans.  Distance  from  the  centre  on  the  axis  of  X,  =41f  inch-   . 
es ;  on  the  axis  of  Y,  =  25  inches. 


THE  ELLIPSE. 


115 


Ex.  2.  Find  the  angle  which  the  tangent  line  in  the  preced- 
ing example  makes  with  the  axis  of  X.  Ans.  149°  3'. 

Ex.  3.  On  an  ellipse  whose  two  axes  are  60  and  40  inches, 
find  the  point  from  which  a  tangent  line  must  be  drawn  in  or- 
der that  it  may  make  an  angle  of  35°  with  the  axis  of  X. 

Ex.  4.  Find  the  equations  of  the  two  lines  which  touch  the 
ellipse  25y''+16ic'=400,  and  which  make  an  angle  of  135°  with 
the  axis  of  X.  '  Ans.  y=-x±  Vil. 

138.  To  find  the  equation  to  the  normal  at  any  point  of  an 
ellipse. 

The  equation  to  a  straight  line 
passing  through  the  point  P,  whose 
co-ordinates  are  x' ^  y'  (Art.  38),  is 
y-y'z:^m{x^x')\  (1) 
and,  since  the  normal  is  perpendic- 
ular to  the  tangent,  we  shall  have 
(Art.  45)  1 

m= 7. 

—m 

But  we  have  found  for  the  tangent  line.  Art.  137, 


Hence 


m  =— ■ 


m- 


aY 
aY 

'h'x" 


Substituting  this  value  in  equation  (1),  we  shall  have  for  the 
equation  of  the  normal  line 

y-y' =%{'«-<»'),  (2) 

where  x  and  y  are  the  general  co-ordinates  of  the  normal  line, 
and  x\  y'  the  co-ordinates  of  the  point  of  intersection  with  the 
ellipse. 


139.  Points  of  intersection  with  the  axes.  To  find  the  point 
in  which  the  normal  cuts  the  major  axis,  make  y=0  in  equa- 
tion (2),  and  we  have,  after  reduction, 

ON,  or  x=z — ^—x\ 


,^<i  ^A;^^^  ^    ^..Jl,    (3 


tt;. 


L^K 


116 


ANALYTICAL   GEOMETEY. 


If  we  subtract  this  value  from  CR,  wliicTi  is  represented  by  x\ 
we  shall  have  the  subnormal 


NR: 


To  find  the  point  in  which  the  normal  cuts  the  minor  axis, 
make  a?=0  in  equation  (2),  and  we  have 


QW=y^ 


y 


C^\ 


140.  Distance  from  the  focus  to  the  foot  of  the  normal.  If 
we  put  e^  for  — 5—  (Art.  127),  we  shall  have 

If  to  this  we  add  F^C,  which  equals  c  or  ae  (Art.  127),  we  have 

F'lSr =ae-\-  e^x'  =  e(a + ex'), 
which  is  the  distance  from  the  focus  to  the  foot  of  the  normal. 

Ex.  1.  In  an  ellipse  whose  major  axis  is  50  inches,  the  ab- 
scissa of  a  certain  point  is  15  inches,  and  the  ordinate  16  inch- 
es, the  origin  being  at  the  centre.  Determine  where  the  nor- 
mal line  passing  through  this  point  meets  the  two  axes. 

Ans.  Distance  from  the  centre  on  the  axis  of  X,  =  5-|  inch- 
es ;  on  the  axis  of  Y,  =9  inches. 

Ex.  2.  Find  the  point  on  the  curve  of  an  ellipse  whose  two 
axes  are  50  and  40  inches,  from  which,  if  an  ordinate  and  nor- 
mal be  drawn,  they  will  form  with  the  major  axis  a  triangle 
whose  area  is  80  inches. 


141.  The  7iormal  at  any  point  of  an  ellijpse  bisects  the  an- 
gle formed  by  Vmes  drawn  from  that  point  to  the  foci. 

Let  FT  be  a  tangent  to  an  el- 
lipse, and  PF,  PF'  be  lines  drawn 
from  the  point  of  contact  to  the 
foci.  Draw  PN  bisecting  the  an- 
gle FPF'.  Then,  by  Geom.,  Bk. 
IV.,Pr.l7, 

FP:FT:;F]Sr:r'N"; 
or  by  composition, 


THE   ELLIPSE. 


117 


FP + FT :  FF^ : :  FT :  F'K  (1) 

But  FP+FT=2^. 

Also  ¥¥'  =  2G=2ae  (Art.  127), 

and  YT=a-\-  ex  (Art.  128). 

Making  these  substitutions  in  proportion  (1),  we  have 

Hence  Y'^=e{a-{-  ex). 

But  by  Art.  140,  e{(i-^ex)  represents  the  distance  from  the  fo- 
cus F'  to  the  foot  of  the  normal.  Hence  the  line  PN,  which 
bisects  the  angle  FPF^,  is  the  normal. 

142.  The  radii  vector es  are  equally  inclined  to  tJte  tangent. 
Since  PN  is  perpendicular  to  TT',  and  the  angle  FPN  is  equal 
to  the  angle  FTN,  therefore  the  angle  FPT  is  equal  to  the  an- 
gle FTT'. 


143.  Second  method  of  drawing  a  tangent  line  to  an  ellijpse. 

Let  P  be  the  point  through 
which  the  tangent  line  is  to  be 
drawn.  Draw  the  radii  vectores 
PF,  PF' ;  produce  PF^  to  G, 
making  PG  equal  to  PF,  and 
draw  FG.  Draw  PT  perpendic- 
ular to  FG,  and  it  will  be  the 
tangent  required;  for  the  angle 
FPT  equals  the  angle  GPT,  which  equals  the  vertical  angle 
FTT^ 


144.  Every  diameter  of  an  ellipse 
Let  PP^  be  a  straight  line  drawn 
through  the  centre  of  the  ellipse, 
and  terminated  on  both  sides  by  the 
curve ;  it  will  be  divided  into  two 
equal  parts  at  the  point  C.  Let  x\ 
y'  be  the  co-ordinates  of  the  point  P, 
and  x'\  y"  those  of  the  point  P'. 


is  bisected  at  the  centre. 


118 


ANALYTICAL    GEOMETEY. 


Since  the  points  P  and  P^  are  on  the  curve,  we  shall  have 
(Art.  121) 

y''~{a^^x'\ 


and 


whence,  by  division, 


y"^^-{a^^x"y, 


y"'-d^-x"'' 


But,  since  the  right-angled  triangles  CPE,  CP^P^  are  similar, 
we  have 

y'    ^' 


y"-x"' 

TT^^^„ 

x''      a'-x'' 

Hence 

x"^-a'-x"" 

Clearing  of  fractions, 

we  obtain 

x"'=zx"'\ 

whence  also  we  have 

y'^=y"\ 

Consequently, 

^-+2/-=aj-+y-, 

or 
that  is. 


CP^CP^'*; 
CP=CF; 


that  is,  PP^  is  bisected  in  C. 


145.  Tangents  to  an  ellipse  at  the  extremities  of  a  diameter 
are  parallel  to  each  other. 

In  Art.  135  we  found  QT~, 

a"  ^ 

and  similarly  CT^=:-77,  where  x' 

rjt  X 

represents  CR,  the  abscissa  of 
the  point  P,  and  x"  represents 
CR',  the  abscissa  of  the  point  P^  But  we  have  found  (Art. 
144)  that  x'  =  x" ;  hence  CT  =  CT^  The  two  triangles  CPT, 
CPT',  have  therefore  two  sides,  and  the  included  angle  of  the 
one  equal  to  two  sides  and  the  included  angle  of  the  other ; 
hence  the  angle  CPT  =.  the  angle  CPT',  and  PT  is  parallel  to 
V'T, 


THE   ELLIPSE.  119 

Hence,  if  tangents  are  drawn  through  the  vertices  of  any 
two  diameters,  they  will  form  a  parallelogram  circumscribing 
the  ellipse. 

r 

O  i4A  "146.  If  from  any  point  in  the  curve  ^chords  are  drawn  to 
the  extremities  of  the  major  axis,  the  jproduct  of  the  tangents 
of  the  angles  which  they  form  with  it,  on  the  same  side,  is 

^" 
equal  to  ——^. 
a 

Let  PA,  PA^  be  two  chords 
drawn  from  the  same  point,  P, 
on  the  ellipse  to  the  extremities 
of  the  major  axis. 

The  equation  of  the  line  PA,  j^j 
passing  through  the  point  A, 
whose  co-ordinates  are  x'=a, 
2^'=0(Art.38),is 

y—m{x—d). 

The  equation  of  PA',  passing  through  the  point  A',  whose 
co-ordinates  are  x''  =  —a,  y'^  =  0,  is 

yzumXx+a). 

At  the  point  of  intersection,  P,  these  equations  are  simulta- 
neous, and,  combining  them  together,  we  have 

f=mm\x'-a').  (1) 

But,  since  the  point  P  is  on  the  curve,  w^e  must  have  at  the 
same  time 

y'~{a'-x')=-^,{x'-a').  (2) 

Comparing  equations  (1)  and  (2),  we  see  that 

mm=^—z. 
a^ 

where  m  denotes  the  tangent  of  the  angle  PAX,  and  m'  de- 
notes the  tangent  of  the  angle  PA'X. 

5^'      147.  Definition.    Two  chords  drawn  from  any  point  in  the 
1/  curve  to  the  extremities  of  a  diameter  are  called  sujpjpleinenta- 
•  ry  chords. 


120  ANALYTICAL   GEOMETRY. 

148.  Bupjplementary  chords  in  a  circle.  A  circle  may  be 
considered  as  an  ellipse  whose  two  axes  are  equal  to  each  oth- 
er ;  hence,  in  a  circle, 

mm'=— 1, 
which  shows  that  the  supplementary  chords  are  perpendicular 
to  each  other  (Art.  4-6). 

149.  If  through  one  extremity  of  the  major  axis  a  chord  he 
drawn  parallel  to  a  tangent  line  to  the  curve ^  the  sujpj^lement- 
ary  chord  will  be  parallel  to  the  diameter  drawn  through  the 
point  of  contact,  and  conversely. 

Let  DT  be  a  tangent  to  the 
ellipse  at  the  point  D,  and  let 
the  chord  AP  be  drawn  parallel 

^j^X y     "^^^  "Hv^    to  it ;  then  will  the  supplement- 

^  ary  chord  AT  be    parallel  to 
the  diameter  DD^,  which  passes 
through  the  point  of  contact,  D. 
Let  x\  y'  designate  the  co-ordinates  of  D.     The  equation  of 
the  line  CD  (Art.  31)  gives 

y'  —  7n'x\ 

whence  tn' ——,. 

X 

But  the  tangent  of  the  angle  which  the  tangent  line  makes 
with  the  major  axis  (Art.  137)  is 

ox 

Multiplying  together  the  values  of  m  and  m\  we  obtain 

mm'=—i. 
a^ 

which  represents  the  product  of  the  tangents  of  the  angles 
which  the  lines  CD  and  DT  make  with  the  major  axis  pro- 
duced. 

But,  by  Art.  146,  the  product  of  the  tangents  of  the  angles 

V 
PAT  and  PAT  is  also  equal  to 5.     Hence,  if  AP  is  paral- 
lel to  DT,  AT  will  be  parallel  to  CD,  and  conversely. 


THE   ELLIPSE.  121 

150.  Definition.  Two  diameters  of  an  ellipse  are  said  to  be 
conjugate  to  one  another  when  each  is  parallel  to  a  tangent 
line  draw^n  through  the  vertex  of  the  other. 

151.  Projperty  of  conjugate  diameters.  Let  DD'  be  any  di- 
ameter of  an  ellipse,  and 
DT  the  tangent  line  drawn 
through  its  vertex,  D,  and 
let  the  chord  AP  be  drawn 
parallel  to  DT;  then,  by 
Art.  149,  the  supplementa- 
ry chord  A^P  is  parallel  to 
DD^  Let  another  tangent,  ET',  be  drawn  parallel  to  AT  ;  it 
will  also  be  parallel  to  DD^  Let  the  diameter  EE^  be  drawn 
through  the  point  of  contact,  E ;  then,  by  Art.  149,  AT  being 
parallel  to  TT,  the  supplementary  chord  AP,  and  also  its  par- 
allel DT,  will  be  parallel  to  EE^  Hence  each  of  the  diameters 
DD',  EE'  is  parallel  to  a  tangent  drawn  through  the  vertex 
of  the  other,  and  by  definition  (Art.  150)  they  are  conjugate  to 
one  another. 

Since  the  conjugate  diameters  DD',  EE'  are  parallel  to  the 
supplementary  chords  AT,  AP,  by  Art.  146,  thejproduct  of  the 
tangents  of  the  angles  which  conjugate  diameters  form  with 

I" 
the  major  axis  is  equal  to  —  — . 

a 

Ex.  1.  In  an  ellipse  whose  axes  are  10  and  8,  a  chord  drawn 
from  one  extremity  of  the  major  axis  forms  with  that  axis  an 
angle  whose  tangent  is  2 ;  what  angle  does  the  supplementary 
chord  form  ?  Aiis. 

Ex.  2.  In  an  ellipse  whose  axes  are  12  and  8,  a  chord  forms 
with  the  major  axis  an  angle  whose  tangent  is  —  3 ;  what  angle 
does  the  supplementary  chord  form  ?  Ans. 

Ex.  3.  In  an  ellipse  whose  axes  are  10  and  8,  find  the  angles 
which  supplementary  chords  drawn  from  the  point  x=l  form 
with  the  major  axis.  Ans. 

Ex.  4.  In  an  ellipse  whose  axes  are  10  and  30,  two  conjugate 

F 


122 


ANALYTICAL    GEOMETRY. 


diameters  are  equally  inclined  to  the  major  axis, 
angle  between  the  two  diameters. 


Find  the 


r 


152.  To  determine  the  co-ordinates  ofthejpoints  ofintersec- 
^vHion  of  a  straight  line  with  an  elli/pse, 

'        Let  the  equation  to  the  ellipse  be 

ay-V¥x'^a'h\  (1) 

and  the  equation  to  a  straight  line  be 

y—mx  +  c.  (2) 

If  this  line  intersects  the  ellipse,  then  we  may  regard  (1)  and 
(2)  as  simultaneous  equations  containing  but  two  unknown 
quantities.     By  substitution  in  equation  (1)  we  obtain 

(a'm'  +  h")x''  +  2a'cmx = {b'  -  c')a% 
the  roots  of  which  equation  give  the  abscissas  of  the  points 
where  the  straight  line  meets  the  curve,  and  the  ordinates  may 
be  found  from  equation  (2).  Hence,  if  the  roots  be  real,  the 
straight  line  will  cut  the  ellipse  in  two  points,  and  it  can  not 
cut  the  ellipse  in  more  than  tw^o  points.  If  tlie  roots  are  equal, 
the  points  of  section  coincide,  and  the  line  is  then  a  tangent. 
If  the  roots  are  imaginary,  the  line  falls  entirely  without  the 
ellipse. 

Ex.  1.  Find  the  co-ordinates  of  the 
points  in  which  tlie  ellipse  w'hose  equa- 
tion is  25?/''4-16aj^  =  400  is  intersected 
by  the  line  whose  equation  isy=2x—5. 
Ans.  a?=+3.Y999,or  +0.5104; 
2/= +2.5998,  or  -3.9792. 
Ex.  2.  Find  the  co-ordinates  of  the 
points  in  which  the  ellipse  whose  equa- 
tion is  4:92/''+36a?''  =  lY64  is  intersected  by  the  line  whose  equa- 
tion is  y=Sx—7,  and  draw  a  figure  representing  the  several 
quantities. 

153.  To  determine  the  co-ordinates  of  the  points  of  inter- 
section of  a  circle  and  ellipse. 

If  the  centre  of  the  circle  is  not  restricted  in  position,  there 


THE   ELLIPSE.  123 

may  be  four  points  of  intersection  corresponding  to  an  equa- 
tion of  the  fourth  degree.     If,  however,  the  centre  of  the  cir- 
cle is  at  one  extremity  of  the  major  axis,  there  will  be  but  two 
points  of  intersection,  w^hich  may  be  easily  found. 
Let  the  equation  to  the  ellipse  be 

and  the  equation  to  the  circle  be 

then,  by  substitution,  we  obtain 

1/ 
r"^ —x'*= —J^ax — x^ ), 

where  x  will  be  found  to  have  two  values,  but  one  of  them  is 
negative,  and  gives  imaginary  values  for  y.  There  will,  there- 
fore, be  but  two  points  of  intersection,  both  having  the  same 
abscissa,  and  the  ordinates  will  differ  only  in  sign. 

Ex.  1.  Find  the  co-ordinates  of  the  points  in  which  the  el- 
lipse whose  equation  is  if —\%{\S)x—x^)  is  intersected  by  the 
circle  whose  equation  is  x^-^-y"  —  ^^. 

Ex.  2.  Find  the  co-ordinates  of  the  points  in  which  the  el- 
lipse whose  equation  is  y" =%%{\^x—x')  is  intersected  by  the 
circle  whose  equation  is  x^^y''  —  V)^. 

If  the  centre  of  the  circle  is  upon  either  axis  of  the  ellipse, 
there  may  be  four  points  of  intersection. 

Ex.  3.  Find  the  co-ordinates  of  the  points  where  the  ellipse 
y''rz:y2_5_(100— a?"")  is  intcrsccted by  the  circles  x^-\-y^-=^^^:\  if-\- 
(^_2yr=64,   2/'4-(aJ-8)^  =  64,  and2/''-f(aj-20/=64. 

The  first  circle  cuts  the  ellipse  in  four  points,  the  second  cuts 
it  in  three  points,  the  third  in  two  points,  and  the  fourth  does 
not  cut  the  ellipse. 

Ex.  4.  Draw  a  figure  representing  these  curves  and  their  in- 
tei-sections. 

154.  Having  given  the  co-ordinates  of  one  extremity  of  a 
diameter^  to  find  those  of  either  extremity  of  the  diameter  corv- 
jugate  to  it. 


124 


ANALYTICAL    GEOMETRY. 


Let  AA',  BB'  be  the  axes  of  an  el- 
lipse; DD',  EE^  a  pair  of  conjugate 
diameters.     Let  x\  y'  be  the  co-ordi- 
^  nates  of  D ;  then  the  equation  to  CD 
(Art.  40)  is  y' 


^     x' 


(1) 


Since  the  conjugate  diameter  EE'  is  parallel  to  the  tangent  at 
D,  the  equation  to  EE'  (Art.  149)  is 

2/—^.^.  (2) 

To  determine  the  co-ordinates  of  E  and  E^,  we  must  combine 
the  equation  to  EE'  with  the  equation  to  the  ellipse,  rt^y  +  JV 

Substituting  the  value  of  y  from  equation  (2),  we  have 

^^^x'-\-l''x'=a%\ 

{h'x''+aY')x'=ay%       iy   i-     -f  < 
a'b''x'=ay'', 

X    -       ^,     , 


Therefore 
or 

whence 


and 


X=±: 


ay 


Taking  the  minus  sign,  in  which  case  x  is  G!N",  and  combining 
with  equation  (2),  we  have 

y_ — -EK 
^      a 

"We  thus  find  the  co-ordinates  of  the  point  E.     The  co-ordinates 

of  the  point  E'  have  the  same  values  with  contrary  signs. 

(^  AJA  ^^^*  "^^^^  ^^^^  of  the  squares  of  any  tioo  conjugate  diame- 
Vy     Urs  is  equal  to  the  sum  of  the  squares  of  the  axes. 

Let  x\  y'  be  the  co-ordinates  of  D ;  then,  by  Art.  154, 


THE   ELLIPSE. 


125 


156.  The  rectangle  contained  ly  the  focal  distances  of  any 
^oint  on  the  dlijpse  is  equal  to  the  square  of  half  the  corre- 
s^onding  conjugate  diameter. 

Let  DD^j  EE'  be  a  pair  of  conjugate 
diameters,  and  from  D  draw  lines  to  the 
foci,  F  and  F^.  Eepresent  the  co-ordi- 
nates of  D  referred  to  rectangular  axes 

Then,  since  CD'  +  CE'=«'-f  J'  (Art. 
155),  we  have 


■y'\ 


f--,- 


^^''-^V^  (Art.  127),    ^"^ 
=DFxDF'(Art.l28); 
that  is,  the  product  of  the  focal  distances  DF,  DF'  is  equal  to 
the  square  of  half  EE^,  which  is  the  diameter  conjugate  to  the 
diameter  which  passes  through  the  point  D. 


t^ 


157.  The  jparallelogr  am  formed  hy  drawing  tangents  through 
the  vertices  of  two  conjugate  diameters  is  equal  to  the  rectan- 
gle of  the  axes. 

Let  DD',  EE'  be  two  con- 
jugate diameters,  and  let  D 
ED'E^  be  a  parallelogram 
formed  by  drawling  tangents 
to  the  ellipse  through  the 
extremities  of  these  diame- 
ters ;  the  area  of  the  paral- 
lelogram is  equal  to  AA'  x  BB'. 

Draw  DM  perpendicular  to  EE',  and  let  the  co-ordinates  of 
D  referred  to  rectangular  axes  be  x',  y' . 

The  area  of  the  parallelogram  DED'E'  is  equal  to  4CE .  DM, 


126  ANALYTICAL   GEOlSIETEr. 

which  is  equal  to  4CE.CT  sin.  CTG,  which  is  equal  to  4CT. 
'EN,  because  EC  and  DT  are  parallel. 

But      CT.--  (Art.  135),  and  EN=—  (Art.  154) ; 

hence  the  parallelogram  DED  'E' = 4 .  - .  —  =:  4a^  ==  A  A'  x  BB^ 

Ex.  1.  In  an  ellipse  whose  axes  are  10  and  8,  what  is  the 
length  of  a  diameter  which  makes  an  angle  of  45°  with  the 
axis  of  X  f     What  is  the  length  of  its  conjugate  ? 

Ans. 

Ex.  2.  What  is  the  altitude  of  the  circumscribed  parallelo- 
gram whose  sides  are  parallel  to  the  conjugate  diameters  of  the 
preceding  example  ?  Ans. 

158.  Equation  to  the  ellipse  referred  to  a  pair  of  conjugate 
diameters  as  axes. 

Let  CD,  CE  be  two  conjugate  semi- 
diameters  ;  take  CD  as  the  new  axis  of 
X,  CE  as  that  of  y;  let  DCA^a,  EC  A 
=  j3.  Let  X,  y  be  the  co-ordinates  of 
any  point  of  the  ellipse  referred  to  the 
original  axes,  and  x\  y'  the  co-ordinates 
of  the  same  point  referred  to  the  new  axes. 

The  equation  of  the  ellipse  referred  to  its  centre  and  axes 
(Art.  121)  is  a\f  -f  J V  =  a'l\ 

In  order  to  pass  from  rectangular  to  oblique  co-ordinates,  the 
origin  remaining  the  same,  we  must  substitute  for  x  and  y  in 
the  equation  of  the  curve  (Art.  56)  the  values 
x=x'  COS.  a-\-y'  COS.  /3, 
y^x'  sin.  a-\-y'  sin.  /3. 
Squaring  these  values  of  x  and  y,  and  substituting  in  the  equa- 
tion of  the  ellipse,  we  have 

x'\a'-  sin.  ^a^l"  cos.  ''a)-Vy'\d^  sin.  ^^■\-h''  cos.  '/3)-f 
2x'y\a^  sin.  a  sin.  j3-|-5''  cos.  a  cos.  f^)  =  a''h% 
which  is  the  equation  of  tlie  ellipse  when  the  oblique  co-ordi- 
nates make  any  angles  a,  ft  with  the  major  axis. 


THE   ELLIPSE.  127 

But,  since  CD,  CE  are  conjugate  semidiameters,  we  must 
have  (Art.  151) 

'ium'  —  tang,  a  tang.  j3=  — — , 

whence  a^  tang,  a  tang.  j3  + J^  =  0. 

Multiplying  by  cos.  a  cos.  /3,  remembering  that  cos.  a  tang,  a 

=sin.  a,  we  have 

o^  sin.  a  sin.  /3H-Z>*  cos.  a  cos.  /3  =  0. 
Hence  the  term  containing  x'y'  vanishes,  and  the  equation  be- 
comes 

x'\a?  sin.  "a  +  V  cos.  ^^a)  +  y'\a?  sin.  ^/3  + 1'  cos.  '/3) = ^''^>',  (1) 
which  is  the  equation  of  the  ellipse  referred  to  conjugate  diam- 
eters. 

If  in  this  equation  we  suppose  y'  —  0,  we  shall  have 

a  sni.  a-\-b   cos.  a 
This  is  the  value  of  CD^,  which  we  shall  denote  by  a'^. 
If  we  suppose  i3?'  =  0,  we  shall  have 

^  ~^^sin.=^/3  +  6''cos.=^j3' 
This  is  the  value  of  CE'',  which  we  shall  denote  by  V^. 

Dividing  equation  (1)  by  a^V^  and  then  substituting  for  the 

coefficients  oix'^  and y'"^  the  equal  values  -75  and  T7i,  we  have  for 

the  equation  to  the  ellipse  referred  to  conjugate  diameters 

x""    y"     , 

or,  suppressing  the  accents  of  the  variables,  we  have 

,  ^  159.  The  square  of  any  diameter  is  to  the  square  of  its  con- 
jugate, as  the  rectangle  of  the  parts  into  which  it  is  divided  hy 
any  ordinate,  is  to  the  square  of  that  ordinate. 

The  equation  of  the  ellipse  referred  to  conjugate  diameters 
may  be  put  under  the  form 

a'Y  =  y\ci^'"-^")'     " 


128 


ANALYTICAL   GEOMETRY. 


This  equation  may  be  reduced  to  tlie 
jy   proportion 


a'':h'-,:a"-x'':y\ 


or    {2ay :  {2by : :  {a'+x){a'-^x) :  y\ 
E"ow  2a'  and  2Z>'  represent  tlie  conju- 
e'     gate  diameters  DD^,  EE' ;  and,  since  x 
represents  CR,  a'  -\-x  will  represent  D'R,  and(x'— a?  will  repre- 
sent DR;  also  PR  represents  y;  lience 

DD'^:EE^=::DRxRD':PR^ 
If  we  draw  a  second  ordinate  P'R'  to  the  diameter  DD',  we 
shall  have 

PR^ :  DR  X  RD^ : :  h"  -.a"::  VTJ' :  DR^  x  R'D^, 
or  PR^ :  PT.^^ : :  DR  X  RD^ :  DR^  x  R^D' ; 

that  is,  the  squares  of  any  two  ordinates  to  the  same  diameter 
are  as  the  products  of  the  jparts  into  which  they  divide  that 
diameter. 


FR/^  But 


160.  To  find  the  polar  equation  to  the  elli/pse,  the  pole  heing 

at  one  of  the  foci. 

1.  Let  F  be  the  pole. 

Let  YV=:r;  angle  PFA  =  0;  then 

^p      rR=:7*  cos.  Q, 

By  AYt.l2S,r=za-€x. 

iZj=CR=,Cr+rR, 

=ae-\-r  cos.  d. 

Therefore  r=ct—ae^—er  cos.  Q. 

T{l  +  eQo^.Q)=a{l-e'), 

a{l-e') 
f> —  — ^^ — 

l-{-e  COS.  0^ 

which  is  the  required  equation  when  6  is  measured  from  the 
radius  to  the  nearer  vertex. 
2.  Let  F'  be  the  pole. 

Let  FTz=:/;  PF'A=:0' ;  then  F^R=/  cos.  9'. 
By  Art.  128,  r'=a-f-ex. 

But  i»=CR=F'R~F^C, 

=r'  COS.  O'—ae. 


•THE   ELLIPSE.  129 


Therefore  r' —a-\-eT'  cos.  O'—ae". 

Hence  q^\1  -  e  cos.  Q')  =  a{i  -  e% 


or  r  =: 


1  —  e  COS.  9 


wliicli  is  the  required  equation  when  0^  is  measured  from  the 
radius  to  the  remote  vertex. 

Ex.  1.  The  axes  of  an  ellipse  are  50  and  40  inches,  and  the 
radius  vector  is  12  inches.     Determine  the  value  of  9, 

Ans.  56°  15'. 

Ex.  2.  The  axes  of  an  ellipse  are  50  and  40  inches,  and  9  is 
equal  to  36°.     Determine  the  radius  vector.  / 

Ans.  10.771  inches. 

Ex.  3.  In  an  ellipse  whose  major  axis  is  50  inches,  the  radius 
vector  is  12  inches,  and  9  is  36°.  Determine  the  minor  axis 
of  the  ellipse.  Ans.  41.67  inches. 

161.  Any  chord  which  passes  through  the  focus  of  an  ellvps6 
is  a  third  jproportional  to  the  majov  axis  and  the  diaiaeter 
^parallel  to  that  chord. 

Let  PP'  be  a  chord  of  an  ellipse 
passing  through  the  focus  F,  and  let     /^  /        N^p 

DD'  be  a  diameter  parallel  to  PP^    ^, 

By  Art.  160,  PF^T^^-^fi-T^ 
^  '  1+^cos.  0 

To  find  the  value  of  FP',  we  must 

substitute  for  9, 180° +  0,  and  we  obtain 

1  —  e  cos.  9 

Hence  PF^^^/^  M^^^'>  ^^. 

1—e  COS.  9  ^  ^ 

But,  by  Art.  158,  ^,j. 

CD'= 


d'Bm.'9-\-b'cos.'W 

a'  sm.'9-{-{a'-a''e')  cos.'9  ^^^^'  -^^^^^ 
a'b' 


a'-a'e'  cos.'9' 
F2 


^ 


I 


130  -  ANALYTICAL    GEOMETRY. 

~l-e'  cos.'O' 
Comparing  equations  (1)  and  (2),  we  find 

2CD^     4CD^ 
^  a     ~   2a    ' 

that  is,  AA':DD'::DD^:PP', 

or  PP'  is  a  third  proportional  to  AA^  and  DD'. 


(2) 


162.  Definition.  The  parameter  of  any  diameter  is  a  third 
proportional  to  that  diameter  and  its  conjugate. 

The  parameter  of  the  major  axis  is  called  the  principal  pa- 

rameter,  or  latus  rectum,  and  its  value  is  —  (Art.  12 G).     The 

.    .    2<2'  ^ 

parameter  of  the  minor  axis  is  -j-.     The  latus  rectum  is  the 

double  ordinate  to  the  major  axis  passing  through  the  focus 
(Art.  126).  Now,  since  any  focal  chord  is  a  third  proportional 
to  the  major  axis  and  the  diameter  parallel  to  that  chord,  and 
since  the  major  axis  is  greater  than  any  other  diameter,  it  fol- 
lows that  the  major  axis  is  the  only  diameter  whose  parameter 
is  equal  to  the  double  ordinate  passing  through  the  focus. 

163.  Definition.  The  directrix  of  an  ellipse  is  a  straight 
line  perpendicular  to  the  major  axis  produced,  and  intersecting 
it  in  the  same  point  with  the  tangent  drawn  through  one  ex- 
tremity of  the  latus  rectum. 

Thus,  if  LT  be  a  tangent  drawn  through  one  extremity  of 
the  latus  rectum  LL',  meeting  the  major  axis  produced  in  T, 
and  NT  be  drawn  through  the  point  of  intersection  perpendic- 
ular to  the  axis,  it  will  be  the  directrix  of  the  ellipse. 

The  ellipse  has  two  directrices,  one  corresponding  to  the  fo- 
cus F,  and  the  other  to  the  focus  F'. 

r 

</164.  TA^  distance  of  any  point  in  an  ellipse  from  either fo- 
eus  is  to  its  distance  from  the  corresponding  directrix,  as  the 
eccentricity  is  to  unity. 


THE   ELLIPSE. 


131 


Let  F  be  one  focus  of  an 
ellipse,  NT  the  corresponding 
directrix ;  F'  the  other  focus, 
and  K'T'  the  corresponding 
directrix.  Let  P  be  any  point 
on  the  ellipse ;  a?,  y  its  co-or- 
dinates, the  centre  being  the  origin.  Join  PF,  PF',  and  draw 
NPN'  parallel  to  the  major  axis,  and  PE  perpendicular  to  it. 


■ht' 

^--■ 

_^ 

p 

t' 

(     F^ 

\ 

L 

Ci 

^  fy 

V^ 

^ 

y 

i 

By  Art.  135, 
Hence,  subtracting  CR  or  a?. 


c     e 


e  e 

But,  by  Art.  128,       r=V¥=a-ex. 
Hence  ^ .  ET,  or  e .  PN = PF 

or,  PF:PN::6:L 

In  like  manner,  we  find  that 

PF^iPN'::^:!. 


165.  To  find  the  area  of  an  ellijpse. 

On  A  A',  the  major  axis  of  an  el- 
lipse, let  a  semicircle  be  described, 
and  within  this  semicircle  inscribe  a 
polygon,  AMM'A/.  From  the  ver- 
tices of  this  polygon  draw  ordinates 
to  the  major  axis,  and  join  the  points  in  which  they  intersect 
the  ellipse,  thus  forming  a  polygon  ANN'A',  having  the  same 
number  of  sides. 

Let  Y,  Y',  etc.,  denote  the  ordinates  of  the  points  M,  M',  etc., 

and  let  y,  y\  etc.,  denote  the  ordinates  of  the  points  N,  W^  etc., 

corresponding  to  the  same  abscissas  a?,  a?','^etc. 

Y+Y' 
The  area  of  the  trapezoid  EMM'E'=r — - — {x—x'\ 

y-\-y' 
and  the  area  of  the  trapezoid  W^WW =—^{x^x'). 

Hence    .       EMM'E':ENISr'E'::  Y+Y^j'-fy'. 


132  ANALYTICAL    GEOMETRY. 

But,  by  Art.  132,         Y :  y : :  a :  5; 
also  Y'  '.y'  :\a:h. 

Whence  Y+ Y'  \y+y'  '.:a:h. 

Therefore  EMM'K' :  ENN'R'  wa-.h. 

In  the  same  manner  it  may  be  proved  that  each  of  the  trap- 
ezoids composing  the  polygon  inscribed  in  the  circle,  is  to  the 
corresponding  trapezoid  of  the  polygon  inscribed  in  the  ellipse, 
as  <a^  is  to  ^  /  hence  the  entii-e  polygon  inscribed  in  the  circle  is 
to  the  polygon  inscribed  in  the  ellipse,  as  a  is  to  h :  and  this 
will  be  true  whatever  be  the  number  of  sides  of  the  polygons. 

If  now  the  number  of  sides  be  indefinitely  increased,  the 
areas  of  the  polygons  will  become  equal  to  the  areas  of  the 
semicircle  and  semi-ellipse  respectively,  and  we  shall  have  the 
first  is  to  the  second  as  a  is  to  b;  or,  denoting  the  area  of  the 
circle  by  S,  and  that  of  the  ellipse  by  s,  we  shall  have 

^'.s'.'.a'.h ;  that  is,  5— -S. 

But  the  area  of  a  circle  whose  radius  is  a  is  represented  by 
ira^  \  hence  s  =  TTab; 

or  the  area  of  an  ellipse  is  equal  to  it  times  the  rectangle  de- 
scribed upon  its  semi-axes. 


166.  Since  7ra J  =  -v/tt V^^  =  V W  X  Tr^'Vwe  find  that  the  area 
of  an  ellipse  is  a  mean  proportional  between  the  areas  of  its 
circumscribed  and  inscribed  circles. 

Ex.  1.  Determine  the  area  of  an  ellipse  whose  two  axes  are 
24  and  18  inches. 

Ex.  2.  The  area  of  an  ellipse  is  40  square  inches,  and  the  la- 
tus  rectum  is  4  inches ;  required  the  axes  of  the  ellipse. 

Ex.  3.  The  axes  of  an  ellipse  are  40  and  50 ;  find  the  areas 
of  the  two  parts  into  which  it  is  divided  by  the  latus  rectum. 


THE   HYPEKBOLA. 


133 


SECTIOK  YII. 


THE  HYPEKBOLA. 


167.  An  hyperbola  is  a  plane  curve  traced  out  by  a  point 
which  moves  in  such  a  manner  that  the  difference  of  its  dis- 
tances from  two  fixed  points  is  always  the  same.  The  two 
fixed  points  are  called  thejvci  of  the  hyperbola. 

Thus,  if  r  and  F^  are  two  fixed  points,  and  if  the  point  P 
moves  about  F  in  such  a  manner  that 
the  difference  of  its  distances  from  F 
and  F^  is  always  the  same,  the  point  P 
will  describe  an  hyperbola,  of  which 
F  and  F'  are  the  foci. 

If  the  point  P'  moves  about  F'  in 
such  a  manner  that  PT— PT^  is  al- 
ways equal  to  PF'-PF,  the  point  P' 
will  describe  a  second  portion  of  the  curve  similar  to  the  first. 
The  two  portions  are  called  branches  of  the  hyperbola. 

The  distance  of  the  point  P  from  either  focus  is  called  the 
Jvcal  distance,  or  the  radius  'vector. 


168.  Mechanical  description  of  the  curve.  From  the  defi- 
nition of  an  hyperbola  the  curve  may  be  described  mechanic- 
ally. Take  any  two  points,  as  F  and 
F^  Take  a  ruler  longer  than  the 
distance  FF',  and  fix  one  of  its  ex- 
tremities at  the  point  F'  so  that  the 
ruler  may  be  turned  round  this  point 
in  the  plane  of  the  paper.  Take  a 
thread  shorter  than  the  ruler,  and 
fasten  one  end  of  it  at  F,  and  the 
other  to  the  end  M  of  the  ruler.     Then  move  the  ruler  on  its 


134 


ANALYTICAL   GEOMETRY. 


pivot  at  F',  while  the  thread  is  kept  constantly  stretched  by  a 
pencil  pressed  against  the  ruler ;  the  curve  described  by  the 
point  of  the  pencil  will  be  a  portion  of  an  hyperbola.  For  in 
every  position  of  the  ruler,  the  difference  of  the  distances  from 
the  variable  point  P  to  the  two  fixed  points  F  and  F'  will  al- 
ways be  the  same,  viz.,  the  difference  between  the  length  of  the 
ruler  and  the  length  of  the  thread. 

If  the  ruler  be  turned  and  move  on  the  other  side  of  tlie 
point  F,  the  other  part  of  the  same  branch  may  be  described. 

Also,  if  one  end  of  the  ruler  be  fixed  at  F,  and  that  of  the 
thread  at  F',  the  opposite  branch  of  the  hyperbola  may  be  de- 
scribed. 

169.  The  centre  of  the  hyperbola  is  the  middle  point  of  the 
straight  line  joining  the  foci. 

A  diameter  is  any  straight  line  passing  through  the  centre, 
and  terminated  on  both  sides  by  opposite  branches  of  an  hy- 
perbola. 

The  diameter  which,  when  produced,  passes  through  the  foci, 
is  called  the  transverse  axis. 

The  latus  rectum  is  the  chord  drawn  through  one  of  the  foci 
pei'pendicular  to  the  transverse  axis. 


170.  To  find  the  equation  to  the  hyperhola. 
Let  F  and  F'  be  tlie  foci,  and  draw  the  rectangular  axes  CX, 
CY,  the  origin  C  being  placed  at  the 
middle  of  FF'.  Let  P  be  any  point 
of  the  curve,  and  draw  PR  perpen- 
dicular to  CX. 

Let  2c  denote  FF',  the  constant 
distance  between  the  foci,  and  let  2a 
denote  FT— FP,  the  constant  dif- 
ference of  the  focal  distances.  De- 
note PF  by  r,  PF'  by  r',  and  let  x  and  y  denote  the  co-ordinates 
of  the  point  P. 

Then,  since  FP'-PE'4-RF^=PE=  +  (CIl-CF)', 


A, 


^     6L 


THE    HYPERBOLA.  135 

we  have  r''=y''J^(oc—cf.  (1) 

Also,  PF'» = PE' + RF" = PR'  4-  (CR + CF)' ; 

that  is,  r"z:zy'  +  {x+c)\  (2) 

Adding  equations  (1)  and  (2),  we  obtain 

^''  +  /'"  =  2(2/»+a5'  +  c');  (3) 

and  subtracting  equation  (1)  from  (2),  w^e  obtain 

r''^—r'^  =  4:cx, 
which  may  be  put  under  the  form 

{r' + r){r'—r) = 4:cx.  (4) 

But,  from  the  definition  of  the  hyperbola,  we  have 

r'—r=2a. 
Substituting  this  value  in  equation  (4),  we  obtain 

,  2cx 

r  +^= — . 
a 

Combining  the  last  two  equations,  we  find 

r'=«+f,  (5) 

r=—a+—.  (6) 

Squaring  these  yalues,  and  substituting  them  in  equation  (3), 
we  obtain 

a'+^-^  =  c''-{-x'+y% 
which  may  be  reduced  to  the  form 

which  is  the  equation  to  the  hyperbola. 
If  we  put  h'^  =  c^—a^^  the  equation  becomes 

l'x'-ay  =  a'b%  (8) 

?-!  =  !'  (9) 

which  is  the  equation  to  the  hyperbola  referred  to  its  centre 
and  transverse  axis. 

This  equation  is  sometimes  written 

y'= J^'-«')-  .  (10) 

The  equation  to  the  ellipse  becomes  the  equation  to  the  hy- 


136  ANALYTICAL    GEOMETRY. 

perbola  by  writing  —  V  for  h^ ;  and  we  shall  find  that  the  hy- 
perbola has  many  properties  similar  to  those  of  the  ellipse. 

171.  Points  of  inter sectio^i  with  the  axes.  To  determine 
where  the  curve  cuts  the  axis  of  X,  make  2/=0  in  the  equation 
to  the  hyperbola,  and  we  obtain 

x=d^a=CA.0YQA\ 
which  shows  that  the  curve  cuts  the  axis  of  abscissas  in  two 
points,  A  and  A^,  at  the  same  distance  from  the  origin,  the  one 
being  on  the  right,  and  the  other  on  the  left ;  and,  since  2CA, 
or  A  A',  is  equal  to  2a,  it  follows  that  the  difference  of  the  two 
lines  drawn  from,  any  jpoint  of  an  hy^perhola  to  the  foci,  is 
equal  to  the  transverse  axis. 

If  we  make  a?=0  in  the  equation  of  the  hy]3erbola,  we  ob- 
tain 

which  shows  that  the  hyperbola  does  not  intersect  the  axis  CY. 

172.  If  with  A  or  A^  as  a  centre,  and  a  radius  equal  to  CF, 
we  describe  a  circle  cutting  the  axis  of  y  in  two  points,  B  and 
B^  we  shall  have  ^ 

CB^=BA»-CA'' 
^o'-a'=h''', 
that  is,  5=:CBorCB'. 

The  line  BB'  thus  determined  is  called  the  conjugate  axis  of 
the  hyperbola.  ;  ^       '^  ' /' 

173.  Figure  of  the  hyperbola  determined.  In  equation  (10), 
Art.  170,  let  x  be  numerically  less  than  a  ;  then  the  values  of  y 
are  imaginary;  therefore  no  point  of  the  hyperbola  is  nearer 
the  axis  of  y  than  ±<7-. 

Let  X  be  numerically  greater  than  a  ;  then  for  each  value  of 
X  there  are  two  equal  values  of  y  with  contrary  signs. 

As  X  increases,  the  values  of  y  increase ;  and  when  x  be- 
comes indefinitely  great,  the  value  of  y  becomes  so  likewise. 

The  hyperbola  tliereforc  consists  of  two  opposite  branches, 


THE   IIYPEEBOLA. 


137 


extending  indefinitely  to  the  right  of  A  and  to  the  left  of  A'-, 
and  symmetrically  placed  with  respect  to  the  axis  XCX'. 

174.  Other  points  of  the  curve  determined.  If  we  wish  to 
determine  other  points  of  the  curve,  we  reduce  the  equation 
to  the  form 


y- 


a 


from  which  we  may  compute  the  value  of  y  corresponding  to 
any  assumed  value  of  x. 

Ex.  Trace  the  curve  whose  equation  is  36aj'— 492/' =  1764. 

Solving  the  equation  for  y,  we  have  /,  -^        3  (i?  y'-  d  '- 

6   r-., ^     "^  ^1    - 

If  X  be  assumed  less  than  7,  the  corresponding  value  of  y  is 
imaginary.  If  we  assume  for  x  different  values  from  7  up- 
ward, we  obtain  the  corresponding  values  of  y  as  given  belovv\ 


Whenaj=ll,2/=±  7.27. 
x  =  12,y=ziz  8.35. 
x=il3,y=:±:  9.39. 
x  =  14:,y=dtlO.Sd. 


When  x=  7,y=0, 

x=  8,y=±S.32. 

X-  9,?/=  ±4.85. 

aj:r=10,2/=±6.12. 
When  x=7,  y=0,  which  gives 
the  point  A.  When  x=z8,  y=± 
3.32,  which  gives  two  points,  cc  and 
a\  one  above  and  the  other  below 
the  axis  of  X;  when  a?=9,  ?/=r± 
4.85,  which  gives  the  points  h  and 
1)'  *  when  x=10,y  =  =t6.12,  which 
gives  the  points  c  and  c',  etc. 

If  we  ascribe  to  a?  a  negative  value,  we  shall  obtain  for  y  the 
same  pair  of  values  as  when  we  ascribed  to  x  the  correspond- 
ing positive  value.  Hence  the  portion  of  the  curve  to  the  left 
of  the  axis  of  Y  is  similar  to  the  portion  to  the  right  of  it. 
Moreover,  there  is  no  point  of  the  curve  between  the  values 
aj=:-f  7  and  x=—7. 


13S 


ANALYTICAL    GEOMETRY. 


175.  To  find  the  foci  of  an  hyjperhola  when  the  two  axes  are 
given.     Since  J'^^c'^—ts^^j  we  have 

that  is,  the  distance  from  the  centime  to 
either  focus  of  an  hyperbola  is  equal  to 
the  distance  between  the  extremities  of  its 


axes. 


If,  then,  from  tlie  centre  C,  with  a  ra- 
dius equal  to  the  diagonal  of  the  rect- 
angle upon  the  semiaxes,  we  describe  an  arc  cutting  the  trans- 
verse axis  produced  in  F  and  F',  the  two  points  of  intersection 
will  be  the  foci  of  the  hyperbola. 


176.  To  find  the  length  of  the  latus  rectum.    According  to 
Art.  170,  eq.  10, 

fj^lx^-a^).         . 


Suppose  a?=c  or  CF ;  then 

y'=-,{c'-a^). 


\^^^ 


But  G^—a^=:¥,  Art.  172 ;  hence  we  have 

or  a'.h'.h:  ?/, 

and  2a\2b\'.2h'.2y. 

But  2y  here  represents  the  double  ordinate  drawn  through  the 
focus,  and  is  called  the  latus  rectum,  Art.  169 ;  hence  the  latus 
rectum  of  any  hyjperhola  is  a  third  jorojportional  to  the  trans- 
verse and  conjugate  axes.  ^,    , 


177.  Equation  to  the  hyjperhola  in  terms  of  the  eccentricity. 
The  fraction  - ,  which  represents  the  ratio  of  CF  to  CA,  or  the 

distance  from  the  centre  to  either  focus  divided  by  half  the 
transverse  axis,  is  called  the  eccentricity  of  the  hyperbola.  If 
we  represent  the  eccentricity  by  ^,  then 


THE    HYPERBOLA. 


139 


—  e^  or  c=ae. 


But  we  have  seen  that 
hence 


or 


Making  this  substitution,  the  equation  of  the  hyperbola  becomes 

wliich  is  the  equation  in  terms  of  the  eccentricity. 

178.  To  find  the  distance  of  any  jpoint  on  the  curve  from 
either  focus.    Equations  (5)  and  (6)  of  Art.  170  are 

ex 

cx 

r=—a-\-—. 

c  ^ 

Substituting  e  for  -,  these  equations  become 

r^=ex-\-a, 
r=ex—a, 
which  equations  represent  the  distance  of  any  point  on  an  hy- 
perbola from  either  focus. 

Multiplying  these  values  together,  we  obtain        ^  ^J 


v-*'t,. 


..^^^ 


which  is  the  value  of  the  product  of  the  focal  distances. 

179.  The  conjugate  hyjperhola.  Suppose  an  hyperbola  to  be 
described  whose  foci  F  and  F'  are  at 
the  same  distance  from  the  centre  C 
as  those  of  the  curve  hitherto  de- 
scribed, but  lie  upon  the  axis  CY  in- 
stead of  CX,  and  suppose  the  differ- 
ence of  the  distances  of  any  point  on 
the  new  curve  from  the  two  foci  is 
Ih  instead  of  ^a ;  then,  retaining  the 
same  axes  of  reference  as  before,  we  shall  have  for  the  new 
position  of  F  and  F', 

FP*=:PK'-hKP=PK»-f-(CK-CF)'; 


140  ANALYTICAL    GEOMETEY. 

that  is  7''^=:x^-\-{y—cf, 

Also,  rT'=PE''+r'K'=PK'+(CE+CF)'; 

that  is,  r"'=x''-\-{y-\-c)\ 

Proceeding  as  in  Art.  170,  we  find 

Putting  a'  for  c^—h"^,  the  equation  becomes 

or  y'^-l^'  +  ci"), 

which  is  the  equation  to  the  new  hyperbola. 

In  this  equation,  suppose  x=0,  and  we  have  y=zh'b ;  that  is, 
the  curve  passes  through  the  points  B  and  B',  and  BB'  is  the 
transverse  axis  of  the  new  curve. 

S uppose        2/ = 0,  and  we  have  x=±aV  —^, 
which  shows  that  the  curve  does  not  meet  the  axis  of  X,  and 
A  A'  is  the  conjugate  axis  of  the  new  curve  (Arts.  171  and  172). 

This  new  hyperbola  is  called  the  liyperbola  conjugate  to  the 
former.  One  hyperbola  is  therefore  said  to  be  conjugate  to  an- 
other, when  the  transverse  and  conjugate  axes  of  the  one  hyjper- 
hola  are  the  conjugate  and  transverse  axes  of  the  other  hyjperhola. 

If  ./J^l^-a^) 

be  the  equation  of  any  hyperbola,  then 

is  the  equation  to  the  hyperbola  conjugate  to  the  former.  The 
latter  equation  may  be  deduced  from  the  former  by  writing 
—a^  for  a",  and  —b"^  for  V^, 

Ex.  1.  Trace  the  curve  whose  equation  is  3a?''— 5?/' =  15. 

Ex.  2.  In  a  given  hyperbola  half  the  difference  of  the  focal 
distances  is  7,  and  half  the  distance  between  the  foci  is  9;  what 
is  the  equation  to  the  hyperbola  ? 

Ex.  3.  "What  is  the  eccentricity  of  the  hyperbola  whose  equa- 
tion is  9a?'-16y'=:144? 

Ex.  4.  What  is  the  equation  of  an  hyperbola  whose  conjugate 
axis  is  6  and  the  eccentricity  1^  ? 


THE   IIYPEKBOLA. 


141 


180.  To  find  the  equation  of  the  hyperbola  when  the  origin 
is  at  the  vertex  of  the  transverse  axis. 
The  equation  of  the  hyperbola  when 


the  origin  is  at  the  centre  is 


(1) 


If  the  origin  is  placed  at  A,  the 
ordinates  will  have  the  same  valne 
as  when  the  origin  was  at  the  centre, 
but  the  abscissas  will  be  changed. 

If  we  represent  the  abscissas  reckoned  from  A  by  x\  w^e 
shall  have  CE=AE+ AC, 

or  x—x'-\-a. 

Substituting  this  value  of  x  in  equation  (1),  we  have 

which  is  the  equation  of  the  hyperbola  referred  to  the  vertex  A. 

^yfi*i      \%\.  Relation  of  ordinates  to  the  transverse  axis.     If  the 
last  equation  be  resolved  into  a  proportion,  we  shall  have 

Now  ^a  represents  the  transverse  axis  AA';  and  since  x  repre- 
sents AE,  2(3^-1- a?  will  represent  A^E;  therefore  {^a-^x)x  rep- 
resents the  product  of  the  distances  from  the  foot  of  the  ordi- 
nate PE  to  the  vertices  of  the  curve.  Hence  we  have  the 
square  of  any  ordinate  to  the  transverse  axis  of  an  hyperbola^ 
is  to  the  product  of  its  distances  from  the  vertices  of  the  curve, 
as  the  square  of  the  conjugate  axis  is  to  the  square  of  the 
transverse  axis. 

If  we  draw  a  second  ordinate  P^E'  to  the  transverse  axis,  we 
shall  have 

PE' :  AE  X  A'E : :  5' :  a» : :  P^E^^ :  AE^  X  A'E', 
or  PE' :  P^E'' : :  AE  x  A'E :  AE'  x  A'E^ ; 

that  is,  the  squares  of  ordinates  to  the  transverse  axis  of  an 
hyperbola  are  to  each  other  as  the  products  of  the  distances 
from  the  foot  of  each  ordinate  to  the  vertices  of  the  curve. 


142  ANALYTICAL    GEOMETEY. 

182.  The  equilateral  hyjperbola.  When  h  is  made  equal  to 
a^  the  equation  of  the  hyperbola  becomes 

y'^'lax-^x''  (Art.  180), 
or  fz^x^-a'     (Art.  170). 

The  hyperbola  represented  by  these  equations  is  called  equi- 
lateral or  rectangular,  and  is  to  the  common  hyperbola  what 
the  circle  is  to  the  ellipse. 

Ex.  1.  Trace  the  curve  whose  equation  is  y^^x'  —  lQ. 

Ex.  2.  Trace  the  curve  whose  equation  is  y''=x^-\-lQ. 

Ex.  3.  Trace  the  curve  whoso  equation  is  'i/  =  10x-\-x^, 

183.  ToJImI  the  equation  to  the  tangent  at  any  ^oint  of  an 
hyjperhola. 

Let  the  equation  to  the  hyperbola  be  t^y  — 2»V=— a'^'. 

Let  x\  y'  be  the  co-ordinates  of  the  point  on  the  curve  at 
which  the  tangent  is  drawn,  and  x'\  y"  the  co-ordinates  of  an 
adjacent  point  on  the  curve.  The  equation  to  the  secant  line 
passing  through  the  points  x\  y'  and  x" ,  y"  (Art.  40)  is 

But,  since  the  points  x' ,  y'  and  x" ,  y"  are  both  on  the  hyperbola, 
we  must  have  d^ij'' — Vx"^  =  —  a^V, 

and  a'y""-'b''x"'=-a'¥', 

therefore,  by  subtraction, 

a\y"'-y")--h\x"'-x")  =  0, 
y"-y'     h'    x"+x' 
^^  x"-x'-a''y"-\-y" 

Substituting  this  value  in  equation  (1),  the  equation  of  the  se- 
cant line  becomes  7  2    y/iy 

The  secant  will  become  a  tangent  when  the  two  points  coin- 
cide, in  which  case,    x'=x",  and  y'  =^y" . 
Equation  (2)  will  then  become 

y-y'=^^-»\  (3) 

which  is  the  equation  to  a  tangent  at  the  point  a?',  y'. 


THE   HYPERBOLA.  143 

Clearing  this  equation  of  fractions,  and  transposing,  we  ob- 
tain dyy' — yxx'  —  a'y'"'  —  h^x'"^ ; 
hence  a^yy'  —  ly'xx'  —  —  a'h'',  (4) 
which  is  the  simplest  form  of  the  equation  to  the  tangent  line. 

h'^x' 
In  equation  (3),  -^r~f  represents  the  trigonometrical  tangent 
a,  y 

of  the  angle  which  the  tangent  line  makes  with  the  transverse 

axis  of  the  hyperbola. 

184.  Points  where  the  tangent  cuts  the  axes.    To  determine 
the  point  in  which  the  tangent  intersects  the 
axis  of  X,  we  make  2/=0,  which  gives 
b'xx'^a'b'', 

that  is,  i»="-7, 

X 

which  is  equal  to  CT.    Therefore 
CTxCR=CA^ 
If  from  CR  or  x'  we  subtract  CT,  we  shall 
have  the  subtangent 


Y 


■B.T^x'--= 


X  X 

To  determine  the  point  in  which  the  tangent  intersects  the 
axis  of  Y,  we  make  aj=0,  which  gives 

which  is  equal  to  C^.     Therefore  C^x  C/'=CB\ 

Hence  it  follows  that  if  a  tangent  and  ordinate  he  drawn 
from  the  same  point  of  an  hyperbola  ^meeting  either  axis  pro- 
duced^ half  of  that  axis  will  he  a  mean  proportional  between 
the  distances  of  the  two  intersections  from  the  centre. 

Ex.  1.  In  an  hyperbola  whose  transv^irse  axis  is  32  inches, 
the  abscissa  of  a  certain  point  is  26  inches,  and  the  ordinate  18 
inches,  the  origin  being  at  the  centre.  Determine  where  the 
tangent  passing  through  this  point  meets  the  two  axes  produced. 
Ex.  2.  Find  the  angle  which  the  tangent  line  in  the  preced- 
ing example  makes  with  the  axis  of  X. 


144  ANALYTICAL   GEOMETRY. 

185.  To  find  the  equation  to  the  normal  at  any  j^oint  of  an 
hyperbola.  The  equation  to  a  straight  line  passing  through 
the  point  P,  whose  co-ordinates  are  x',  y'  (Art.  38),  is 

y-y'=m{x-x')',  (1) 

and  since  the  normal  is  perpendicular  to  the  tangent,  we  shall 
have  (Art.  46) 

1 

m— -/. 

— m 

But  we  have  found  for  the  tangent  line  (Art.  183) 

,     ¥x' 
m ——ri\ 

,  ^y 

hence  m  —  —n—,, 

o  X 

Substituting  this  value  in  equation  (1),  we  shall  have  for  the 

equation  of  the  normal  line 

where  x  and  y  are  the  general  co-ordinates  of  the  normal  line, 
and  x\  y'  the  co-ordinates  of  the  point  of  intersection  with  the 
hyperbola. 

186.  Points  of  intersection  vnth  the  axes.  To  find  the 
point  in  which  the  normal  cuts  the  trans- 
vei*se  axis,  make  ?/=0  in  equation  (2),  and 
we  have,  after  reduction, 

a'+h'      , 
a 
If  from  CN  we  subtract  CR,  which  is  repre- 
sented by  x\  we  shall  have  the  subnormal 

a  a 

To  find  the  point  in  which  the  normal  cuts  the  axis  of  Y, 
make  i»=0  in  equation  (2),  and  we  have,  after  reduction, 

a'  +  'b'      , 

which  equals  C/i. 


THE   HYPERBOLA.  145 

187.  Distance  from  the  focus  to  the  foot  of  the  normal.  If 
we  put  e^  for  — 3—  (Art.  177),  we  shall  have 

If  to  this  we  add  F'C  (see  next  figure),  which  equals  c  or  a^, 
Art.  177,  we  have 

Y'l^ =ae-\-  e^x' = e{a  -f  ex'), 
which  is  the  distance  from  the  focus  to  the  foot  of  the  normah 
Ex.  In  an  hyperbola  whose  transverse  axis  is  32  inches,  the 
abscissa  of  a  certain  point  is  26  inches,  and  the  ordinate  18 
inches,  the  origin  being  at  the  centre.  Determine  where  the 
normal  line  passing  through  this  point  meets  the  two  axes. 

188.  ^  tangent  to  the  hyperbola  bisects  the  angle  contained 
by  lines  drawn  from  the  point  of  contact  to  the  foci. 

Let  PT  be  a  tangent  line  to  the  hyperbola,  and  PF,  PP'  two 
lines  drawn  from  the  point  of 
contact  to  the  foci ;  then  the  an- 
gle FPT=rTT. 

For  CT=-(Art.l84), 

and      CF=«tf(Art.l77); 

hence  FT =ae——=-(ex—a\ 
X     x^  ^' 

and    ^'T=ae-\ — =-{ex-\-a). 

X       X 

Therefore  ^T\Y'T:\ex-a\ex+a, 

::PF:PF'(Art.l78). 
Hence  PT  bisects  the  angle  FPF'  (Geom,,  Bk.  lY.,  Prop.  17). 

189.  If  FT  be  produced  to  M,  and  the  normal  PN  be  drawn, 
it  will  bisect  the  exterior  angle  FPM.  For,  since  PN  is  per- 
pendicular to  TT',  and  the  angle  FPT  is  equal  to  FTT  or  its 
vertical  angle  MPT',  therefore  the  angle  FPIS'=MP]S';  or  the 
normal  bisects  the  angle  included  by  on^  radius  vector  and 
the  other  produced. 


14:6 


ANALYTICAL   GEOMETRY. 


190.  To  draw  a  tangent  to  the 
hyperbola  through  a  given  jpoint 
of  the  curve.  Let  P  be  the  given 
point ;  draw  the  radii  yectores  PF, 
PF' ;  on  PF^  take  PG  equal  to  PF, 
and  draw  FG.  Draw  PT  perpen- 
dicular to  FG,  and  it  will  be  the 
tangent  required,  for  it  bisects  the 
angle  FPF'. 


191.  Every  diameter  of  an  hyperhola  is  bisected  at  the  cen-- 
tre.  Let  PP'  be  a  straight  line  drawn  through  the  centre  of 
the  hyperbola,  and  terminated  on  both  sides  by  the  two  branch- 
es of  the  curve ;  it  will  be  divided  into  two  equal  parts  at  the 
point  C.  Let  x\  y*  be  the  co-ordinates  of  the  point  P,  and 
x" ^  y"  those  of  the  point  P'. 

Since  the  points  P  and  P'  are  on  the  curve,  we  shall  have 
(ArtirO)  z. 

y'^=-{x'^-a^), 


and 


y 


a^ 


a'); 


whence,  by  division, 

y"      x'^-^a' 
y"'-x"'^a'' 

But,  since  the  right-angled  triangles  CPR,  CP'K'  are  similar, 

we  have  y^     x' 


hence. 


X' 


Clearing  of  fractions,  we  obtain 

x"^x"'', 
w^hence  also  we  have  y'^=y"^. 

Consequently,  x'''-\-y'''=x"''+y' 

or  CF  =  CP'^; 

that  is,  CP=CF; 

that  is,  PP'  is  bisected  in  C. 


THE   HYPEKBOLA. 


147 


192.  Tangents  to  an  hyperbola  at  the  extremities  of  a  diam- 
eter are  parallel  to  each  other. 

Let  PP'  be  a  diameter  of  an  hy- 
perbola, and  let  PT,  V'T  be  tangents 
drawn  through  its  extremities ;  then 
is  PT  parallel  to  PT^ 

In  Art.  184  we  found  QT~,  and 
for  the  same  reason  CT'= -77,  where  a?'  represents  CE,  the  ab- 

X 

scissa  of  the  point  P,  and  x"  represents  CK',  the  abscissa  of  the 
point  P'.  But  in  Art.  191  w^e  have  found  that  x' =x"  \  hence 
CT = CT'.  The  two  triangles  CPT,  CPT'  have  therefore  two 
sides  and  the  included  angle  of  the  one  equal  to  two  sides  and 
the  included  angle  of  the  other;  hence  the  angle  CPT = the 
angle  CP^T',  and  PT  is  parallel  to  VT, 


193,  If  from  the  extremities  of  the  transverse  axis  two  lines 
he  drawn  to  meet  on  the  curve^  the  product  of  the  tangents  of 
the  angles  which  they  form  with  that  axis  on  the  same  side  is 

equal  to  — . 


a 


Let  PA,  PA'  be  two  lines  drawn  from  the  extremities  of 
the  transverse  axis  to  the  same  point 
P  on  the  hyperbola.  The  equation 
of  the  line  PA  passing  through  the 
point  A,  whose  co-ordinates  are  x'  =  a, 
y'  =  0,Art.  38,is 

y=m{x—a). 
The  equation  of  PA'  passing  through 
the  point  A',  whose  co-ordinates  are  aj''=  —  ^,  y''  =  0,  is 

y—m'{x-\-d). 
At  tlie  point  of  intersection,  P,  these  equations  are  simultane- 
ous, and,  combining  them  together,  we  have 

y'^mm'ix^-a'),  (1)    C. 


148  ANALYTICAL    GEOMETRY. 

But,  since  tlie  point  P  is  on  the  curve,  we  must  have  at  the 
same  time 

f~{x'-a'){Avtm).  (2) 

Comparing  equations  (1)  and  (2),  we  see  tliat 

mm  =^, 

where  m  denotes  the  tangent  of  the  angle  PAX,  and  m'  de- 
notes the  tangent  of  the  angle  PA'X. 

194.  Definition.  Two  lines  drawn  from  any  point  on  the 
curve  to  the  extremities  of  a  diameter,  are  called  supplement- 
ary  chords.  , 

195.  Supplementary  chords  in  the  equilateral  hyperhola. 
In  the  equilateral  hyperbola  a=h,  and  we  have 

inm'  =  l, 

1 

or  m  —  —,. 

w^hich  shows  that  the  angles  formed  by  the  supplementary 
chords  with  the  transverse  axis  oii  the  same  side  are  comple- 
mentary to  each  other  (Trig.,  Art.  28). 

196.  If  through  one  extremity  of  the  transverse  axis,  a  chord 
he  drawn  parallel  to  a  tangent  line  to  the  cic7've,  the  supple- 
mentary chord  will  he  parallel  to  the  diameter  drawn  through 
the  point  of  contact^  and  conversely. 

Let  DT  be  a  tangent  to  the  hy- 
perbola at  the  point  D,  and  let  the 
chord  AP  be  drawn  parallel  to  it; 
then  will  the  supplementary  chord 
AT  be  parallel  to  the  diameter 
DD^,  which  passes  through  the 
point  of  contact,  D. 

Let  x\  y'  denote  the  co-ordinates 
of  D.  The  equation  of  the  line 
CD  (Art.  31)  gives  y'  =  m'x' ; 


THE   IIYPEKBOLA. 


149 


whence 


mm 


y 


But  the  tangent  of  the  angle  which  the  tangent  line  makes 
with  the  transverse  axis  (Art.  183)  is 

a'y' 
Multiplying  together  the  values  of  m  and  m\  we  obtain 


mm 


a' 


which  represents  the  product  of  the  tangents  of  the  angles 
which  the  lines  CD  and  DT  make  with  the  transverse  axis. 
But  by  Art.  193  the  product  of  the  tangents  of  the  angles 

PAX,  PA'X  is  also  equal  to  -^.     Hence,  if  AP  be  parallel  to 

DT,  AT  will  be  parallel  to  CD,  and  conversely. 


197.  The  last  Proposition  is  also  true  when  applied  to  a  tan- 
gent to  the  conjugate  hyperbola;  that  h,  if  through  one  ex- 
trem^ity  of  the  transverse  axis  of  an  hyjperhola  a  chord  he 
drawn  jparallel  to  a  tangent  line  to  the  conjugate  hyperbola^ 
the  supplementary  chord  will  he  parallel  to  the  diameter 
drawn  through  the  point  of  contact^  and  conversely. 

Let  ET'  be  a  tangent  to  the  hyperbola  which  is  conjugate  to 
the  former  hyperbola,  and  let  the 
chord  AT  be  drawn  parallel  to  ET', 
and  through  the  point  of  contact,  E, 
let  the  diameter  EE'  be  drawn  ;  then 
will  EE'  be  parallel  to  the  supple- 
mentary chord  AP. 

Let  x" ^  y"  denote  the  co-ordinates 
of  E.  The  equation  of  the  line  CE, 
Art.  31,  gives  y"=mf'x"\ 


whence 


m"  = 


y 


Tlie  equation  of  the  conjugate  hyperbola  (Art.  179)  is 
aY-h'x'^a'h'-, 


150  ANALYTICAL   GEOMETRY. 

and,  proceeding  as  in  Art.  183,  we  shall  find  that  the  tangent 
of  the  angle  ET'C  is 

ay 

Hence  we  have  ram" —-i^. 

which  represents  the  product  of  the  tangents  of  the  angles 
ECA  and  ETA.  But  this  has  been  found  (Art.  196)  equal  to 
the  product  of  the  tangents  of  the  angles  DCX  and  DTX,  or 
PAX.  Hence,  if  AT  be  parallel  to  ET',  AP  will  be  parallel 
to  CE,  and  conversely. 


U^ 


198.  Conjugate  diameters.  Each  of  the  diameters  DD',  EE' 
is  thus  seen  to  be  parallel  to  a  tangent  line  drawn  through  the 
vertex  of  the  other  diameter.  Two  diameters  thus  related  are 
said  to  be  conjugate  to  each  other.  Thus  we  see  that  the  pro- 
duct of  the  tangents  of  the  angles  which  conjugate  diameters 

.  r     7  .    .     .       ,       h' 

form  with  the  transverse  axis  %s  equal  to  —i. 

199.  Of  any  two  conjugate  diameters^  one  m^eets  the  original 
hyperbola,  and  the  other  the  conjugate  hyperbola. 

Let  y—mx  be  the  equation  to  any  diameter,  and  let 
aY-b'x^^-a'b' 
be  the  equation  to  the  hyperbola. 

To  determine  the  points  in  which  the  diameter  intersects  the 
curve,  we  must  combine  these  two  equations,  and  we  have 
{a\a'-b'')x'=-a'b\ 

or  «''=ri 3—3-  ('1) 

0  —am  ^  ^ 

In  like  manner,  for  the  conjugate  hyperbola  we  shall  find 

^  -a'm'-b"  ^"^ 

The  values  of  x  in  equation  (1)  will  be  real  as  long  as  a^m* 
is  less  than  5',  but  imaginary  when  a^m'  is  greater  than  J^  In 
the  former  case  the  diameter  intersects  the  curve,  but  in  the 


tup:  hyperbola. 


151 


latter  it  does  not.  The  values  of  x  in  equation  (2)  are  real 
when  h"^  is  less  than  ci^m^^  but  imaginary  when  h^  is  greater 
than  a'm\ 

Now,  in  the  case  of  conjugate  diameters,  we  have 

mm  —  -^,  or  iribm  =  — . 

Hence,  if  m"  be  less  than  — ,  m''  will  be  greater  than  -^ ;  in  this 
case  the  first  diameter  meets  the  original  hyperbola,  and  the 
second  the  conjugate  hyperbola.  If  m'be  greater  than  — ,  m'' 
will  be  less  than  —^ ;  in  this  case  the  first  diameter  meets  the 

coniugate  hyperbola,  and  the  second  the  original  hyperbola,     y 

C^'f^  /^\<^^     ^   ■-  ':     ^  \  '''^: 

200';  Having  given  the  co-ordinates  of  one  extremity  of  a 

diameter^  to  find  those  of  either  extremity  of  the  diameter 
conjugate  to  it. 

Let  AA',  BB'  be  the  axes  of  an  hy- 
perbola, DD',  EE^  a  pair  of  conjugate 
diameters.  Let  x\  y'  be  the  co-ordi- 
nates of  I) ;  then  the  equation  to  CD 
(Art.  40)  is 

(1) 


V 
y=.—, .  X. 
^     X 


Since  the  conjugate  diameter  EE^  is  parallel  to  the  tangent 
at  D,  the  equation  to  EE'  (Art.  183)  is 

h'x' 


y= 


ay 


X. 


(2) 


To  determine  the  co-ordinates  of  E  and  E',  we  must  combine 
the  equation  to  EE'  with  the  equation  to  the  conjugate  hyper- 
bola aY-h'x^=a'h'  (Art.  179). 

Substituting  the  value  of  y  from  equation  (2),  we  have,  after 
reduction,  (b'^x'^ — a^y"^)x^ — a^y'"^ ; 


whence 


or 


ay' 


152 


ANJlLYTICAL  geometey. 


Also,  from  equation  (2)  we  liave 

hx' 

which  are  the  co-ordinates  of  the  points  E  and  E^  The  ab- 
scissa of  E  is  positive,  and  that  of  E'  negative ;  hence  the  upper 
sign  applies  to  E,  and  the  lower  to  E^ 

201.  The  difference  of  the  squares  of  any  two  conjugate  di- 
ameters is  equal  to  the  difference  of  the  squares  of  the  axes. 
Let  x\  y'  be  the  co-ordinates  of  D  ;  then,  by  Art.  200, 

QJ)^-Q■E^^x'^^y'^-^—A- 
^         0         a 

h'x"-ay    ay-l'x" 

~        b"         '^        a' 

=  a'-l\ 


202.  The  rectangle  contained  hj  the  focal  distances  of  any 
jpoint  on  the  hyperbola  is  equal  to  the  square  of  half  the  cor- 
responding conjugate  diameter, 

LetDD',  EE'be  a  pair  of  conju- 
gate  diameters,  and   from  D  draw 
lines  to  the  foci  F  and  F';  then 
DFxDF^^CE^ 
Kepresent  the   co-ordinates   of  D 
by  x\  y'. 

Then,    since    CD»-CF=a'-J» 
(Art.  201),  we  have 
QY.'^QW-a^-^-V 
^x^-^-y'^-a'-^h' 

=x"'-\--lx"-a')--a'-{-l>' 


={i4y-a^ 


=e'x''-a'{Ait.l17) 
=DFxDF'(Ai't.l78); 
that  is,  the  product  of  the  focal  distances  DF,  DF^  is  equal  to 


THE   nYPERBOLA. 


153 


the  square  of  half  EE',  which  is  the  diameter  conjugate  to  that 
which  passes  through  the  point  D. 

203.  The  j>ci^'cd^^^ogravi  formed  hy  drawing  tangents 
through  the  vertices  of  two  conjugate  diameters  is  equal  to 
the  rectangle  of  the  axes. 

Let  DD',  EE^  be  two  conjugate  diameters,  and  let  DED'E' 
be  a  parallelogram  formed  by  drawing 
tangents  to  the  hyperbola  through  the  ex- 
tremities of  these  diameters  ;  the  area  of 
the  parallelogram  is  equal  to  AA'  x  BB^ 

Draw  DM  perpendicular  to  EE',  and 
let  the  co-ordinates  of  D  be  x\  y' . 

The  area  of  the  parallelogram  DED'E' 

is  equal  to  4CE  .  DM,  which  is  equal  to  4CE  .  CT  sin.  CTII, 

which  is  equal  to  4CT  .  EN,  because  EC  and  DT  are  parallel.'^ 

d  hx' 

But  CT=-  (Art.  184),  and  EN^. —  (Art.  200).     Hence  the 

^  a"    hx'       ^ 

parallelogram  DED'E^=4-  .  —  =  4^^>.=AA'xBB^ 


204.  Equation  to  the  hyj^erhola  referred  to  any  two  conju- 
gate diameters  as  axes. 

Let  CD,  CE  be  two  con j  ugate  semi-diameters ;  take  CD  as 
the  new  axis  of  x,  CE  as  that  of  y ;  let 
DC  A = a,  and  EC  A = j3.  Let  x,  y  be  the 
co-ordinates  of  any  point  of  the  hyperbola 
referred  to  the  original  axes,  and  x\  y'  the 
cordinates  of  the  same  point  referred  to 
the  new  axes. 

The  equation  of  the  liyperbola  referred  to  its  centre  and 
axes  (Art.  170)  is         ahf  -  JV  =  -  a^h\ 

In  order  to  pass  from  rectangular  to  oblique  co-ordinates, 

the  origin  remaining  the  same,  we  must  substitute  for  x  and  y 

in  the  equation  of  the  curve  (Art.  56)  the  values 

x=x'  cos.  a-\-y'  COS.  /3, 

y—x'  sin.  a  +  y'  sin.  /3. 

G2 


154  ANALYTICAL  GEOMETRY. 

Squaring  these  values  of  x  and  ?/,  and  substituting  in  the 
equation  of  the  hyperbola,  we  have 

x'\a^  sin/a-^>'  cos.''a)+?/'X^'  sin//3-Z'=  cos/j3) 
+  2x'y\a'  sin.  a  sin.  ^  —  1/  cos.  a  cos.  j3)  =  —a'b'', 
which  is  the  equation  of  the  hyperbola  when  the  oblique  co- 
ordinates make  any  angles  a,  /3  with  the  transverse  axis. 

But,  since  CD,  CE  are  conjugate  semidiameters,  we  must 

have  (Art.  198)  ,  ^     J' 

mm  =tang.  a  tang.  p=— , 

whence  a^  tang,  a  tang.  j3— J^  =  0. 

Multiplying  by  cos.  a  cos.  j3,  remembering  that  cos.  a  taiig.  a 

=  sin.  a,  v/e  have 

^^'^  sin.  a  sin.  /3  — Z*"  cos.  a  cos.  j3=:0. 
Hence  the  term  containing  x'y'  vanishes,  and  the  equation  be- 
comes 

x'\a'  sin.'a-Z.'  cos. ''a)  +  y' V  sin.^/S-^*'  cos.^j3)  =  -a'^'', 
which  is  the  equation  of  the  hyperbola  referred  to  conjugate 
diameters. 

If  in  this  equation  we  suppose  y'  —  0,  we  shall  have 


;^  Cv, 


This  is  the  value  of  CD',  which  we  shall  denote  by  a'^. 
If  we  suppose  x'  =  0,  we  shall  have  , 

-<^'^'  ..    ' 

Now,  since  we  have  supposed  that  the  new  axis  of  x  meets  the 
curve,  we  know  that  the  new  axis  of  y  will  not  meet  the  curve 
(Art.  199),  so  that  —a'b' 

is  not  2i, positive  quantity.  If  we  denote  it  by  —5^',  the  equa- 
tion to  the  hyperbola  referred  to  conjugate  diameters  will  be 

h'^x'^-ay^za^h"; 
or,  suppressing  the  accents  on  the  variables, 
h"'x'-a"f^a"y\ 

or  Z^-r»  =  l- 


THE    HYPERBOLA. 


155 


205.  The  square  of  any  diameter  of  an  hyperhola  is  to  the 
square  of  its  conjugate,  as  the  rectangle  under  any  two  seg- 
ments of  the  diameter  is  to  the  square  of  the  corresponding 
ordinate. 

Let  DD',  EE'be  two  conjugate  diameters  of  an  hyperbola, 
and  from  any  point  of  the  curve,  as 
P,  let  PR  be  drawn  parallel  to  EC, 
meeting  the  diameter  DD'  produced 
inE. 

The  equation  of  the  hyperbola  re- 
ferred to  conjugate  diameters  may 
be  put  under  the  form 

a'Y  =  h'\x'-a''). 
This  equation  may  be  reduced  to  the  proportion 
a":h"'.\x'-a'''.y\ 

or  {2ay :  (2^0'  - '  («+«0(«-«') '  2/'- 

"Now  2a^  and  2b'  represent  the  conjugate  diameters  DD',EE'; 
and  since  x  represents  CR,  x-\-a'  will  represent  D^R,  and  x—a' 
will  represent  DR ;  also  PR  represents  y.     Hence 
DD^':EE^»::DRxRD':PR^ 
If  we  draw  a  second  ordinate  P'R'  to  the  diameter  DD',  we 
shall  have  PR^ ;  DR  x  RD' : :  b''  \a"\\  P^R^' :  DR'  x  R'D', 
or  PR' :  P'R'" : :  DR  x  RD' :  DR'  x  R'D' ; 

that  is,  the  squares  of  any  two  ordinates  to  the  same  diameter 
are  proportional  to  the  rectangles  under  the  corresponding 
segments  of  the  diameter. 

206.  To  find  the  polar  equation  to  the  hyj^erbola^  the  pole 
being  at  one  of  the  foci.  ^ 

1.  Let  F  be  the  pole.  ^^ 

Let  YY=r;  the  angle  KYV=.^\ 
then  FR=r  cos.  PrR=  —r  cos.  0. 
By  Art.  178, 

rr:^ex—a  ; 
but    .      a?=CR=CF-|-rR 

z=ae—r  cos.  0.  ^ 


156  ANALYTICAL    GEOMETRY. 

Therefore  r=ae'—erQ,o?>.Q—a. 

Hence  r{l-\-e  cos.  B)  =  a{e* — 1), 

l-\-e  cos.  0'  ^  -' 

wliicli  is  the  equation  required. 
2.  Let  F'  be  the  pole. 

Let  FT=/;  angle  VYA^O';  then  F'E=/  cos.  0', 
By  Art.  178,  r'=ex  +  a  / 

but  i»=CE=F^E-FC 

=r^  cos.  Q'—ae. 
Therefore  r'  —  er'  cos.  & — «^^'  -f  <3^. 

Hence  /(I  — ^  cos.  0')  =  <2(1  — 6"^)  =  — ^(6*— 1),     / 

or  ^^'  =  , ^ i„  (2) 

1  —  6  COS.  0'  ^  ^ 

which  is  the  equation  required. 

207.  Form  of  the  hyjper'bola  traced.  The  form  of  the  hy- 
perbola may  be  traced  from  its  polar  equation.  In  equation 
(1),  suppose  0=0;  then  r=a{e—l).  If  we  measure  off  this 
length  on  the  initial  line  from  the  pole  F,  we  shall  obtain  the 
point  A  as  one  of  the  points  of  the  curve. 

While  Q  increases,  l-\-e  cos.  Q  diminishes,  and  r  increases; 

and  when  0  =  90°,  r=—,  which  determines  another  point  of 

the  curve. 

When  d  becomes  greater  than  90°,  cos.  0  becomes  negative, 

and  7"  continues  to  increase  until  l-{-e  cos.  0  =  0, or  cos.  0=  — -, 

^  e^ 

when  r  becomes  infinite.  Thus,  while  0  increases  from  0  until 
COS.  0=—-,  that  portion  of  the  curve  is  traced  out  which  be- 
gins at  A,  and  passes  on  through  P  to  an  indefinite  distance 
from  the  origin. 

When  l-\-e  cos.  0  becomes  negative,  r  becomes  negative,  and 
we  measure  it  in  the  direction  opposite  to  that  in  which  we 
should  measure  it,  if  it  were  positive.     Thus,  while  0  increases 


THE   HYPERBOLA. 


157 


to  180°,  that  portion  of  the  curve  is  traced  out  which  begins 
at  an  indefinite  distance  from  C  in  the  lower  left-hand  quad- 
rant, and  passes  through  Q  to  A^ 

As  9  increases  from  180°,  r  continues  negative,  and  increases 
numerically  until  l-{-e  cos.  0  again  becomes  zero.  Thus  the 
branch  of  the  curve  is  traced  out  which  begins  at  A',  and  pass- 
es on  through  Q'  to  an  indefinitely  great  distance  from  C. 

As  0  continues  to  increase,  the  value  of  1.+  6  cos.  6  again  be- 
comes positive ;  r  is  again  positive,  and  is  at  first  indefinitely 
great,  and  then  diminishes.  Thus  the  portion  of  the  curve  is 
traced  out  which  begins  at  an  indefinitely  great  distance  from 
C  in  the  lower  right-hand  quadrant,  and  passes  on  through  P' 
to  A.  Thus  both  branches  of  the  hyperbola  are  traced  out  by 
one  complete  revolution  of  the  radius  vector. 


208.  A7iy  cho7'd  which  passes  through  the  focus  of  an  hyper- 
hola  is  a  third  jproportional  to  the  transverse  axis  and  the  di- 
ameter parallel  to  that  chord. 

Let  PP^  be  a  chord  t)f  an  hyperbola  passing  through  the 
focus  F,  and  let  EE^  be  a  diameter  par- 
allel to  PP; 

By  Art.  206,  pp^/|(^'-^) 

*^  '  1  +  6  COS.  Q 

•  To  find  the  value  of  FP^,  we  must  sub- 
stitute for  0, 180°  4-0,  and  we  obtain 

aie'-l) 


Hence 


FP'  = 
PF  = 


1—e  COS.  d' 
2a(e'-l) 


l-e'cos.'6' 
Proceeding  as  in  Art.  161,  we  find  the  value  of  CE'  equal  to 

a%e'-l) 


ei^^  J 


Hence 
that  is. 


PF= 


-e'cos.'O' 
2CE'    4:CE' 


a    ~  ^a   ' 
AA':EE^::EE':PF, 
or  PP'  is  a  third  proportional  to  A  A'  and  EE'. 


^^f 


^c^^ 


158 


ANALYTICAL    GEOMETRY. 


209.  Definition.  The  parameter  of  any  diameter  is  a  third 
proportional  to  that  diameter  and  its  conjugate. 

The  parameter  of  the  transverse  axis  is  called  the  principal 

parameter,  or  latus  rectum^  and  its  value  is  —  (Art.  176).     The 

parameter  of  the  conjugate  axis  is  -r-.     The  latus  rectum  is 

the  double  ordinate  to  the  transverse  axis  passing  througli  the 
focus  (Art.  176).  Now,  since  any  focal  chord  is  a  third  propor- 
tional to  the  transverse  axis  and  the  diameter  parallel  to  that 
chord,  and  since  the  transverse  axis  is  less  than  any  other  di- 
ameter of  the  same  hyperbola,  it  follows  that  the  transverse 
axis  is  the  only  diameter  whose  jparameter  is  equal  to  the 
douhle  ordinate  jpassing  through  the  focus. 

In  the  equilateral  hyperbola  a=h,  and  the  latus  rectum  is 
equal  to  either  of  the  axes  of  the  curve. 

i"-/ 210.  Definition.  The  directrix  of  an  hyperbola  is  a  straight 
line  perpendicular  to  the  transverse  axis,  and  intersecting  it  in 
the  same  point  with  the  tangent  to  the  curve  at  one  extremity 
of  the  latus  rectum. 

Thus,  if  LT  be  a  tangent  drawn  through  one  extremity  of 
the  latus  rectum  LL^,  meeting  the  transverse  axis  in  T,  and  NT 
be  drawn  tlirough  the  point  of  intersection  perpendicular  to 
the  axis,  it  will  be  the  directrix  of  the  hyperbola. 

The  hyperbola  has  two  directrices,  one  corresponding  to  the 
focus  F,  and  the  other  to  the  focus  Y'. 


211.  The  distance  of  any  point  in  an  hyperbola  from  either 
focus  is  to  its  distance  from  the  corre- 
sponding directrix,  -as  the  eccentricity 
is  to  unity. 

Let  F  be  one  focus  of  an  h}^erbola, 
NT  the  corresponding  directrix ;  F^  the 
other  focus,  and  N'T'  the  corresponding 
directrix.     Let  P  be  any  point  on  the 
hyperbola,  x',  y'  its  co-ordinates,  the  origin  being  at  the  centre. 


THE   HYPERBOLA.  159 

Join  PF,  PF',  and  draw  PNN'  parallel  to  the  transverse  axis, 
and  PP  perpendicular  to  it. 

By  Art.  184,  CT=-:=-/ 

c     & 

hence  CK-CT=P]S'=«'-- 


e 
But, by  Art.  178,       r=6a;'-a-PF; 
hence  e.PlS'^PF, 

or  PF:PN::6:1. 

In  like  manner  we  find  that 

PF^:PN'::6:1. 

212.  Conic  sections  compared.  In  Art.  82  the  parabola  was 
defined  to  be  a  curve  every  point  of  which  is  equally  distant 
from  the  focus  and  directrix,  while  in  the  ellipse  and  hyper- 
bola these  distances  have  been  found  to  be  in  the  ratio  of  the 
eccentricity  to  unity.     In  the  ellipse,  the  eccentricity,  being 

c 
equal  to  -  (Art.  127),  is  less  than  unity,  while  in  the  liyperbola 

(Art.  177)  it  is  greater  than  unity.  In  each  of  these  curves  the 
two  distances  have  to  each  other  a  constant  ratio.  In  the  par- 
abola this  ratio  is  unity,  in  the  ellipse  it  is  less  than  unity,  while 
in  the  hyperbola  it  is  greater  than  unity.  These  curves,  being 
the  sections  of  a  cone  made  by  a  plane  in  different  positions, 
are  called  the  conic  sections  ^'  so  that  a  conic  section  may  be 
defined  to  be  a  curve  traced  out  by  a  point  wliich  moves  in 
such  a  manner  that  its  distance  from  a  fixed  point  bears  a  con- 
stant ratio  to  its  distance  from  a  fixed  straight  line.  If  this 
ratio  be  unity,  the  curve  is  called  a  parabola ;  if  less  than  unity, 
an  ellipse ;  and  if  greater  than  unity,  an  hyperbola ;  and  all 
the  properties  of  these  curves  may  be  deduced  from  this  defi- 
nition. 


160 


ANALYTICAL  GEOMETRY. 


ON  THE  ASYMPTOTES  OF  THE  HYPERBOLA. 
213.  It  was  shown  in  Art.  199  that  if  a  line  drawn  through 
the  centre  of  an  hyperbola  meets  the  curve,  irC  must  be  less 

7  3  7 

than  — ,  or  inK^- ;  and  if  the  line  meets  the  conjugate  hy- 

^^  ^  7  3  7 

perbola,  Qn'^  must  be  greater  than  -j,  or  m>  ±-. 

Let  AA',  BB'  be  the  two  axes  of  an  hyperbola,  and  through 
p;::  j^  the  vertices  A,  A',  B,B^  let  lines  be  draWn 
perpendicular  to  these  axes ;  and  let  DD', 
P      EE',  the  diagonals  of  the  rectangle  thus 
X  formed,  be  indefinitely  produced. 
Then,  since 


and 


^^^    DA    h 
tang.DCX=^=-, 

^        ^,^,,    E^A        h 
tanff.E'CX: 


AC""     a' 

it  follows  that  the  lines  CD,  CE'  will  never  meet  the  curve  at 
any  finite  distance  from  C. 

The  lines  CD,  CE',  indefinitely  produced,  are  called  asymp- 
totes of  the  hyperbola. 


214.  Definition.  An  asymptote  of  any  curve  is  a  line  which 
continually  approaches  the  curve,  coming  indefinitely  near  to 
it,  but  meets  it  only  at  an  infinite  distance  from  the  origin. 

Since  the  lines  DD'  and  EE^  pass  through  the  centre,  and 
are  inclined  to  the  transverse  axis  at  an  angle  whose  tangent 

=  ±-,  their  equation  will  be 


215.  The  diagonals  of  the  rectangle  formed  hy  lines  drawn 
through  the  extremities  of  the  axes  and  perpendicular  to  the 
axes^  are  asymptotes  to  the  curve^  according  to  the  defiiiition 
of  Art  214. 


THE   HYPERBOLA.  161 

Let  tlie  equation  to  the  hyperbola  (Art.  170)  be 

The  equation  to  the  line  CL,  the  diagonal  of  the  rectangle 
DED'E^is  j^ 

^      a 
Let  MPR  be  an  ordinate  meeting  the  hyperbola  in  P,  and  the 
straight  line  CL  in  M ;  then,  if  CE  be  denoted  by  a?,  we  have 

PE=-v^^^^,   -  V. 

and  MK--. 

Hence  W^=-{x--^/x'-a') 


ah 


x-{-  -^x^—a^ 

If,  then,  the  line  MK  be  supposed  to  move  from  A  parallel 
to  itself,  the  value  of  x  will  continually  increase,  and  the  dis- 
tance MP  will  continually  diminish ;  and  if  we  suppose  the 
point  P  of  the  curve  to  recede  to  an  infinite  distance  from  the 
origin,  MP  will  become  zero.  7 

In  like  manner  the  line  CL',  whose  equation  is  y=— — , 

meets  the  curve  below  the  transverse  axis  at  an  infinite  dis- 
tance from  the  origin. 

216.  Asymjptote  to  the  conjugate  hyperbola.  The  line  CL  is 
also  an  asymptote  to  the  conjugate  hyperbola;  for, let  PE  be 
produced  to  meet  the  conjugate  hyperbola  in  P' ;  then  (Art.  179) 

P'E=-VSM^.     ^^ 
a  I 

Hence    ^^  -  -^      P'Mi^-(v^M^^-a?) 

ab 


■y/x^  -\-a^  +  x 


162  ANALYTICAL   GEOMETKY. 

Therefore,  if  CR  or  x  be  indefinitely  increased,  P^M  will  be 
indefinitely  diminished,  and  hence  CL  is  an  asymptote  to  the 
conjugate  hyperbola. 

217.  An  asymptote  tnay  he  considered  as  a  tangent  to  the 
hyperbola  at  a  point  infinitely  distant  from  the  centre. 

The  equation  to  a  tangent  at  any  point  x\  y'  of  the  curve 

(Art.  183)  is  a'yy'  -  h'xx'  =  -  a'b% 

I'xx'     b' 
or  y—-^-f——,  (1) 


Now  y'—zh-  ^Jx"  —  d\ 

^  a 

If  x'  becomes  indefinitely  great,  then  o^  vanishes  when  com- 
pared with  a?'",  and  we  have         7 

^  a 

Substituting  this  value  in  equation  (1),  the  equation  to  the 
tangent,  when  the  point  x\  y'  is  infinitely  distant,  becomes 

Vxx'      a      a¥ 
^~~   a^       bx'~bx' 

bx    ah 

=  ±—±-7. 
a     X 

But  when  x'  is  infinite,         —}=0; 

bx 
hence  yz=±—, 

which  is  the  equation  to  the  asymptote  (Art.  214).  Hence  the 
asymptote  is  a  tangent  to  the  curve  at  a  point  infinitely  distant 
from  the  centre. 

218.  The  asymptotes  are  the  diagonals  of  every  parallelo- 
gram formed  by  drawing  tangents  through  the  vertices  of  two 
conjugate  diameters. 

Let  DED'E'  be  a  parallelogram  formed  by  drawing  tangents 
to  the  hyperbola  through  the  vertices  of  two  conjugate  diam- 
eters DD',  EE';  the  diagonals  Tt,Tt'  will  be  asymptotes  of 
the  curve. 


THE   HYPERBOLA.  163 

Let  x\  y'  be  the  co-ordinates  of  tlie  point  D ;  then  the  co- 
ordinates of  E,  the  extremity  of  the 
conjugate  diameter  (Art.  200),  are 

ay'       _  hx' 

-7-  and  — . 

0  a 

Draw  the  diagonal  DE,  and  it  will 

bisect  CT  in  N  (Geom.,  Bk.  I.,  Prop.  33). 

The  co-ordinates  of  K  are 


Hence  we  have 


,(...f)  .„d  i{,y,'^). 


Ix' 


tang.  NCX = .,,  which  equals  -. 

But  -  IS  the  tangent  of  the  angle  which  the  asymptote  makes 

with  the  transverse  axis  (Art.  214) ;  hence  CT  coincides  with 

one  of  the  asymptotes. 

Also,  since  the  diagonal  DE  passes  through  the  points 

,     ,       ^ay'    hx' 
aj,2/,and-^,  — , 

the  tangent  of  the  angle  which  it  makes  with  the  transverse 
axis  (Art.  40)  is  ^^/ 

^,,  which  equals  — -. 

b  ,  ^ 

But  ——is  the  tangent  of  the  angle  which  the  other  asymptote 

makes  with  the  transverse  axis ;  hence  DE  is  parallel  to  the 
other  asymptote.  And  since  DT'E'C  is  a  parallelogram,  DT^ 
=E'C,  which  equals  EC;  and  since  DT'  is  parallel  to  EC, ED 
is  parallel  to  CT^     Hence  T't'  is  the  other  asymptote. 

219.  Hence  we  see  that  the  line  joining  the  extremities  of 
two  conjugate  diameters  is  jparallel  to  one  asymptote,  and  is 
bisected  hy  the  other. 


164  ANALYTICAL   GEOMETRY. 

Also,  if  a  tangent  line  he  drawn  at  any  poiiit  of  an  hyper- 
bola, the  part  included  between  the  asymptotes  is  equal  to  the 
parallel  diameter. 

Moreover,  if  x  and  y  are  the  co-ordinates  of  any  point  on  the 
asymptote  referred  to  two  conjugate  diameters,  then  we  shall 
have  y.xwb'  \a\ 

b'x 

which  is  therefore  the  equation  to  the  asymptote  referred  to  a 
pair  of  conjugate  diameters. 

220.  If  any  chord  of  the  hyperbola  be  produced  to  meet  the 
asymptotes,  the  parts  included  between  the  curve  and  the  as- 
ymptotes will  be  equal. 

Let  PP^  be  any  chord  of  the  hyperbola,  and  let  it  be  pro- 
duced to  meet  the  asymptotes  in  M  and  M' ; 
then  will  PM  be  equal  to  P'M^ 

Draw  CY,  the  semidiameter  to  the  conju- 
gate hyperbola,  parallel  to  PP',  and  draw  CX 
conjugate  to  CY ;  then  PP^  is  a  double  ordi- 
nate to  CX,  and  is  bisected  in  P. 

The  equation  to  the  hyperbola  referred  to 
CX,  CY  (Art.  204)  is 

y=J-,^/'^F^:^%  (1) 

and  the  equation  to  the  asymptotes  (Art.  219)  is 

Now  to  the  same  abscissa  CK  there  correspond  (from  eq.  1) 
two  equal  ordinates  with  opposite  signs ;  hence  we  have 

PP=P^K. 
Also,  from  eq.  2,  MR = M'R. 

Therefore,  by  subtraction,  MP=MT', 
as  was  to  be  proved. 

If  the  tangent  line  TT'  be  drawn  parallel  to  MM',  the  trian- 
gles CTT',  CMM'  will  be  similar ;  and  since  MR  is  equal  to 


THE   HYPERBOLA. 


165 


M'KjNT  will  be  equal  to  NT';  that  is,  the  jportion  of  a  tan- 
gent included  between  the  asymptotes  is  bisected  at  the  jpoint 
of  contact. 

221.  If  a  straight  line  he  drawn  through  any  point  on  an 
hyperbola^  the  rectangle  of  the  parts  intercepted  between  that 
point  and  the  asymptotes^  will  be  equal  to  the  square  of  the 
parallel  semidiameter. 

Let  a  straight  line  drawn  through  the  point  P  on  the  hyper- 
bola meet  the  asymptotes  in  M  and  M';  then  we  have 
PM .  PM'  =  (MR-PK)(ME4-  PR) 
z=MR'-PR^ 


But 

and 

hence 
that  is, 


MR' 


MR'=— ,a;'  (Art.  219), 

VwJ^^lx'^a")  (Art.  204) ; 

-VW-J-^lx'-x'  +  a")  =  b"', 
a^  / 


PM.PM'-J^ 

or,  the  rectangle  of  the  parts  PM  and  PM'  is  equal  to  the 
square  of  the  parallel  semidiameter. 

222.  To  find  the  equation  to  the  hyperbola  referred  to  the 
asymptotes  as  axes. 

Let  CX,  CY  be  the  original  axes  coinciding  with  tlie  axes  of 
the  hyperbola,  and  let  CD,  CE  be  the 
new  axes,  inclined  to  CX  on  opposite 
sides  of  it  at  an  angle  /3,  such  that 

tang.  j3  =  -  (Art.  213).    Let  a?,  y  be  the 

co-ordinates  of  a  point  P  referred  to  the 
old  axes,  and  x' ^  y'  the  co-ordinates  of 
the  same  point  referred  to  the  new  axes. 

The  formulas  for  passing  from  rectangular  to  oblique  co-or- 
dinates, the  origin  remaining  the  same  (Art.  56),  are 

x—x'  cos.  a  -f  2/'  COS.  ^, 

2/=a?' sin.  a  +  y' sin./B. 


166  ANALYTICAL   GEOMETRY. 

Butj  since  a=  —  j3,  these  equations  become 
x—{Qc!-\-y')  C0S./3, 
2/=(2/'— aj')sin./3. 

Now  sin.  /3 = -7^,  and  cos.  )3 = -^ ; 

also,  CP  =  CA^  +  AP=:a'+5». 

Represent  CLbjc/  then 

sin.  /3  =  -,  and  cos.  j3 = -. 
0  c 

Therefore  x = — ^-^,  and  y = -^ -, 

Substitute  these  values  in  the  equation  to  the  hyperbola, 

and  we  have 

a'hXx'  -yy^a'bXx'  ■\-yy  =  -a'hY, 
or  (x^^yy^{x'+yy=^c'; 

that  is,  4a7y  =  c'; 

or,  suppressing  the  accents, 

^'    ^'  +  ^'     -  Aaa> 

which  is  the  equation  of  the  hyperbola  referred  to  the  asymp- 
totes as  axes. 

223.  Equation  to  the  conjugate  hyjperhola.  The  equation 
to  the  conjugate  hyperbola  referred  to  the  same  axes  may  be 
found  by  writing  —a^  for  ^',  and  —  ^'  for  h^  (Art.  179).  We 
shall  then  have 

xy=—^,    -^   -   \V. 

In  the  case  of  an  equilateral  hyperbola,  the  angle  DCE  =  90° ; 
that  is,  the  asymptotes  are  perpendicular  to  each  other.  For 
all  other  hyperbolas  the  asymptotes  make  oblique  angles  with 
each  other. 

Ex.1.  Trace  the  curve  whose  equation  referred  to  rectangu- 
lar axes  is  xy=10. 


THE   HYPERBOLA. 


167 


x=2,y=z 

6. 

aj=3,y= 

3.33. 

aj=4,y= 

2.5. 

x=o,y= 

2. 

{2j=6,2/= 

1.66. 

We  may  assume  any  value  for  x,  and  the  corresponding  value 
of  y  may  be  found  from  the  equation.     Thus,  if 

i»=l,2/=10.  x=  7,y=1.43. 

x=  8,y=1.25. 

x^  9,2/==l.ll. 

ajr=10,2/=1.00. 

a;=ll,2/=0.91. 

a;=12,2/=0.83. 
These  values  determine  the  points  of  the  curves  a^  h,  c,  d,  etc. 
If  X  is  negative,  y  is  also  negative, 
and  the  points  a\  h\  c\  etc.,  will  be 
determined  in  the  third  quadrant. 
As  X  increases  indefinitely,  y  de- 
creases, and  the  curve  is  unlimited 
in  the  direction  of  x  positive,  but 
continually  approaches  the  axis  of 
X  without  actually  reaching  it.  The 
same  is  true  for  the  direction  of  x 
negative,  and  for  each  direction  of  the  axis  of  y. 

Ex.  2.  Trace  the  curve  whose  equation  referred  to  oblique 
axes  is  xy=—10. 


^ 


■ 


224.  Parallelogram  on  any  abscissa  and  ordinate.  .  Let  P 
be  any  point  on  the  hyperbola,  from  which  draw 
PM,P]S"  parallel  to  the  asymptotes,  and  repre-  -  mhp 
sent  these  co-ordinates  by  x,  y ;  then,  by  Art. 

If  we  multiply  each  member  of  this  equation  by  sin.  2j3,  we 
shall  have  i  \k  i 

xy  sin.  2/3=^^t^  sin.  2/3,  (1) 

where  2/3  is  the  angle  included  by  the  asymptotes.     The  first 
member  of  this  equation  represents 

CMxCNxsin.MCK  (2) 

But  ON  X  sin.  MCN  is  the  perpendicular  from  N  upon  the  line 


k 


ca 


^ui-\^c^r 


168  ANALYTICAL   GEOMETRY. 

CM ;  lience  expression  (2)  represents  the  area  of  the  parallelo- 
gram CNPM. 

Since    sin.  2/3  =  2  sin.  /3  cos.  /3 = ^r^^  (Art.  222), 

ah           5-^  .  ^  ^ 
the  second  member  of  eq.  1  reduces  to  -^.    :.     *■ '^^"'^"^ 

Hence  the  parallelogram  CNPM  described  on  the  abscissa 
and  ordinate  of  any  point  on  the  curve,  is  equal  to  half  the 
rectangle  under  the  semiaxes,  or  one  eighth  the  rectangle  under 
the  axes. 

225.  To  find  the  equation  to  the  tangent  at  any  jpoint  of  an 
hyperbola  when  the  curve  is  referred  to  its  asymptotes  as  a^es. 

Let  x',  y'  be  the  co-ordinates  of  the  point  of  contact,  x" ,  y" 
the  co-ordinates  of  an  adjacent  point  on  the  curve. 

The  equation  to  the  secant  line  passing  through  these  points  is 


y-y'=^-^k^-^').  (1) 

Since  the  two  given  points  are  on  the  hyperbola,  we  have 
(Art.  222) 

^  x'y' 

Hence  x'y'-x"y",  or  y"=-^. 

x'n' 
Therefore  y"-y'=.-^-y' 


X 


~~  x"^  "~^)> 


whence 


y  -y       y_ 

x"-x'-^x" 
^By  substitution,  eq.  (1)  becomes 


y-y'=-'^k«-^')-  (2) 

If  we  suppose  x' =x'\  and  y' —y'\  the  secant  will  become  a 
tangent,  and  equation  (2)  will  become 


THE   HYPEEBOLA.  169 


which  is  tlie  equation  to  the  tangent  line. 

If  we  clear  this  equation  of  fractions,  we  shall  have 
yx'  -  x'y'  =-xy'-\-  x'y' ; 

therefore  yx'  -^-xy'  —  "^x'y'  =  — - — , 

which  is  the  simplest  form  of  the  equation  to  the  tangent  line. 


226,  Points  of  intersection  with  the  axes.     To  find  where 
the  tangent  at  x\  y'  meets  the  axis  of  ab- 
scissas, put  y—^iw,  the  equation  to  the  tan- 
gent line,  and  we  have 

xy'^'lx'y', 
or  x  —  '^xf; 

that  is,  the  abscissa  CT'  of  the  point  where     i 
tlie  tangent   meets   the   asymptote   CE  is 
double  the  abscissa  CR  of  the  point  of  tangency. 

To  find  where  the  tangent  cuts  the  axis  of  Y,put  aj=0  in  the 
equation  to  the  tangent  line,  and  we  have 

or  y  —  '^y'' 

that  is,  CT  is  double  of  PR. 

Also,  because  PR  is  parallel  to  CT,  TT'  is  double  of  PT,  or 
the  tangent  TT'  is  bisected  in  P ;  that  is,  if  a  tangent  line  he 
drawn  at  any  point  of  an  hyperbola,  the  part  intercepted  be- 
tween the  asymptotes  is  bisected  at  the  point  of  contact 

H 


170  ANALYTICAL   GEOMETEY. 


SECTION  YIII. 

GENEEAL  EQUATION  OF   THE    SECOND   DEGEEE. 

227.  We  have  seen  that  the  equations  of  the  circle,  the  par- 
abola, ellipse,  and  hyperbola  are  all  of  the  second  degree ;  we 
will  now  inquire  whether  any  other  curve  is  included  in  the 
general  equation  of  the  second  degree. 

The  general  equation  of  the  second  degree  between  two  va- 
riables may  be  written 

ax'-{-hxy^cy''-{-dx-\-e'y-{-f=0,  (i; 

which  contains  the  first  and  second  powers  of  each  variable, 
their  product,  and  an  absolute  term. 

We  shall  suppose  the  axes  to  be  rectangular ;  for  if  they  were 
oblique  we  might  transform  the  equation  to  one  referred  to 
rectangular  axes,  and  we  should  obtain  an  equation  of  the  same 
degree  as  the  above,  and  which  could  not,  therefore,  be  more 
general  than  the  one  we  have  assumed. 

228.  To  remove  certain  terms  from  the  general  equation. 
We  wish,  if  possible,  to  cause  certain  terms  of  this  equation  to 
disappear.  For  this  purpose  we  may  change  both  the  origin 
and  direction  of  the  co-ordinate  axes,  without  assigning  any 
particular  values  to  the  quantities  which  determine  the  position 
of  the  new  axes.  By  this  means,  indeterminate  quantities  are 
introduced  into  the  transformed  equation,  to  which  such  values 
can  afterwards  be  assigned  as  will  cause  certain  of  its  terms  to 
vanish.  Instead  of  changing  botli  the  origin  and  direction  of 
the  co-ordinate  axes  at  once,  it  is  more  convenient  to  effect 
these  changes  successively. 

229.  The  terms  containing  the  first  powers  ofx  and  y  in 
the  equation  of  the  second  degree^  may  in  general  he  mode  to 
disajppear  hy  changing  the  origin  of  the  co-ordinates. 


GENERAL   EQUATION   OF  THE    SECOND   DEGREE.  171 

In  order  to  effect  this  transformation,  substitute  for  x  and  2/ 
in  equation  (1)  the  vahies 

x=x'+h, 

by  which  we  pass  from  one  system  of  axes  to  another  system 
parallel  to  the  first  (Art.  54). 
The  result  of  this  substitution  is 

ax"  +  bxy  +  cy''  +  (2a/i  ^lh'Vd)x' -\-  i^ch  -\-hh  +  e)y' 
+  ah^  +  hhh + ch'  +  dh  +  eh  +/=  0. 
Now,  in  order  that  the  terms  involving  the  first  powers  of  x' 
and  y'  may  disappear,  we  must  have 

^ah-^-hh  +  d^O, 
and  2c7ii-\-hh  +  e=zO. 

From  these  equations  we  obtain 

7     '^cd-he  2ae-bd  0,^^^.yM^^ 

These  are  the  values  of  h  and  h  which  render  the  proposed 
transformation  possible ;  hence,  denoting  the  constant  quantity 

ah^-\-lhk^cU^dh^eh^rf 
\>yf',  the  transformed  equation  becomes 

ax'' -^rhx'y'  +  cy'^ +/'  =  ().  (2) 

"When  ^'— 4f<^c=0,  the  above  values  of  h  and  Jc  become  infi- 
nitely great,  and  the  proposed  transformation  is  impossible. 

If  equation  (2)  is  satisfied  by  any  values  x^,  y^  of  the  varia- 
bles, it  is  also  satisfied  by  the  values  —.'??,,  —y^  Hence  the  new 
origin  of  co-ordinates  is  the  centre  of  the  curve  represented  by 
equation  (1). 

Thus,  if  b''—Aiac  be  not  =0,  the  curve  represented  by  (''.)  has 
a  centre,  and  its  co-ordinates  are  h  and  h,  the  values  of  which 
are  given  above. 

We  may  suppress  the  accents  on  the  variables  in  equation 
(2),  and  write  it       ax'  +  hxy^  +  cy'  -\-J'  =  0.  '  (3) 

230.  The  term  coyitaining  xy  in  the  general  equation  of  the 
second  degree  may  he  taken  away  by  changing  the  directions 

.  »X-  %''  ■'■■' 


172  ANALYTICAL   GEOMETRY. 

For  this  purpose  put 

x—x'  COS.  ^—y'  sin.  0, 
y—x'  sin.  ^-\-y'  cos.  0. 
Substituting  these  values  of  x  and  y  in  equation  (3),  and  ar- 
ranging the  result,  we  have 

x'^{a  cos.  ''^  +  c  sin.  "0  +  J  sin.  0  cos.  0) 
-\-y'''{a  sin.'^  +  c  cos.'0— J  sin.  0  cos.  0) 
+a;yi2(c-«)  sin.  0  cos.  04-%os.''6/-sin.='0)}+/'  =  O.  (4) 
Now,  in  order  that  the  term  involving  x'y'  may  become  zero, 
we  must  have 

2(c— fl^)  sin.  0  COS.  0+%os.''0— sin.'0)=O. 
But  by  Trig.,  Art.  73, 

2  sin.  0  COS.  0 = sin.  20 ;  also  cos.  ^0 — sin.  ''0 = cos.  20 ; 
hence  {c—d)  sin.  20  +  ^  cos.  20=0, 


h     (h 


or  tang.  20  = .    C^^^tA^.'       (5) 


a— c 


^'tvi 


Since  the  tangent  of  an  angle  may  have  any  magnitude  from 
zero  to  infinity,  this  value  of  tang.  20  is  always  possible,  what- 
ever be  the  values  of  a,  ^,  and  c ;  hence  such  a  value  of  0  may 
always  be  found  as  shall  remove  the  term  involving  x'y'  from 
equation  (4),  and  the  general  equation  is  reduced  to  the  form 

or,  suppressing  the  accents  on  the  variables,  we  have 

K^V&it^f=^.'Y^P  (6) 

By  solving  this  equation  we  have  ^  • ,  .,  ,/ 


from  which  we  see  that  if  A,  B,  and/*'  have  the  same  sign,  the 
quantity  under  the  radical  is  negative,  and  equation  (6)  repre- 
sents an  imaginary  curve. 

If  A  and  B  have  the  same  sign,  andy^  has  the  contrary  sign, 
the  equation  represents  an  ellipse  (Art.  121). 

If  A  and  B  have  different  signs,  the  equation  represents  an 
hyperbola  (Art.  170). 

If  A=B,  the  equation  represents  a  circle  (Art.  60).     ^ 

lfy^  =  0,  and  A  and  B  have  the  8a7ne  sign,  the  equation  can 


GENEEAL   EQUATION   OF  THE   SECOND   DEGREE.  173 

only  be  satisfied  by  the  values  x—0  and  y=0;  that  is,  the 
equation  represents  a  point,  viz.,  the  origin. 

If  y'=:0,  and  A  and  B  have  diferent  signs,  equation  (6)  re- 
duces to  y=  ±33 Y  4- T^, 
which  represents  two  intersecting  straight  lines. 

231.  To  find  the  values  of  the  coejffiGients  A  and  B  in  equa- 
tion (6)  in  terms  of  a,  ^,  a7id  c. 

Since         A=a  cos.  d-\-G  sin.  '0 + J  sin.  0  cos.  0, 
and  ^—a  ^m.^'Q+c  cos.'' 0—b  sin.  9  cos.  0, 

we  have,  by  addition,  observing  that  sin.  ''^-f  cos.  ^9  =  1, 

A+B=a-{-c,  (m) 

and  by  subtraction,  observing  that  cos.  ^"0— sin.  ^"0=:  cos.  20, 
A-B=:(a-c)  COS.  29+h  sin.  29. 
E"ow,  since   sec.  =  VTTtang?, 

bye,.  (5),     .ee.20  =  xA;i5.^^^^g±?; 

hence  cos.  2  0  =  --; , 

Vb'^  +  ia-cf 

and  sin.  20—    ,  =. 

V6''-\-{a-cy 

Hence  we  have 


Vd''-\-{a-cy     Vb'+{a-cy 
h'  +  {a-cf 


Adding  and  subtracting  successively  (m)  and  (;^),  we  have 
A^^a^-G^Vb'-^ia-cy], 
B=^j{a-\-c^Vb''-\-{a-cf\. 
Multiplying  together  these  values  of  A  and  B,  we  have 

A.B=^ i ^ --r=z . 

4  4 

Hence  A  and  B  have  the  same  sign  or  different  signs  accord- 
ing as  4ac— 5'  is  positive  or  negative.    ^^-Ij^Cl:^^     ^^    .  i.  ic. 


44    '■^   t^ 


174  ANALYTICAL   GEOMETRY. 

232.  Particular  case  considered.  "We  will  now  consider 
the  case  in  which  lP-—^aG  is  zero.  We  can  not  in  this  case 
destroy  the  terms  involving  x  and  y  by  transferring  the  origin 
to  the  centre  of  the  curve,  as  was  done  w^ith  the  ellipse  and 
liyperbola,  but  we  may  remove  the  term  involving  xy  by  chang- 
ing the  direction  of  the  axes. 
Let  the  equation  be 

ax^-\-lxy-\-ey^^-dx-{-ey-^f—^.  (1) 

Put  x—x'  cos.  Q—y'  sin.  0, 

y—x'm\.Q-\-y'Q.o^.^, 
Substituting  these  values  in  equation  (1),  we  have 
x'''(a  cos.'^^+c  sin.  ^0+ J  sin.  0  cos.  0) 
-Vy'^'ia  sin.^'^  +  c  cos.  ^^—h  sin.  0  cos.  0) 
+ajy{2(c-t?)  sin.  0  cos.  0+J(cos.'O-sin.''0)} 
^x'{d  COS.  0+6  sin.  ^)^y'{e  cos.  ^-d  sin.  0)+/=O.  (2) 
In  order  that  the  term  involving  x'y'  may  become  zero,  we 
must  have  2(c — t?)  sin.  0  cos.  0  +  J(cos.  '0 — sin.  '0)  =  0 ; 

whence,  as  in  Art.  230,  tang.  20= , 

and  the  co-efficients  of  x'""  and  y'",  as  in  Art.,  231,  are 

Wa^c-^^/h'-^ia^cy]. 
One  of  these  coefficients  must  therefore  vanish,  since  their  prod- 

uct  (Art.  231)  is  — j — ,  which,  by  hypothesis,  =0.     Suppose 

the  coefficient  of  x''^  =  0\  if  we  suppress  the  accents  on  the  va- 
riables, equation  (2)  will  assume  the  form 

^  C2/^4.D^-fE2/+/=0.  (3) 

Transposing  and  dividing  by  C,  we  have 

^.  2/  i-  o  -  .    C      0* 

Adding  ^TTa  to  each  member,  we  have 

/       EV        W        E^       f\ 

1?  T)  E'        y^ 

Put  Z=-.— ,  M=— p,  and  ^^^Jfr)— f)?  a^<^  equation  (4) 

may  be  written  {y—lf='M{x—n). 


GENERAL   EQUATION   OF  THE   SECOND   DEGREE.  175 

If  now  the  origin  be  transferred  to  a  point  whose  co-ordi- 
nates  are  a?=n,  y—l^ 

we  shall  have,  by  writing  x-{-n  for  ic,  and  y-\-l  for  3/, 

y=Ma!,  (5) 

which  is  the  equation  to  a  parabola. 

If  in  equation  (3)  D=0,we  have 

E      1      

which  gives  ?/=  "20-20 '^^'-^^f^ 

which  represents  two  parallel  straight  lines,  or  one  straight  line, 
or  an  imaginary  curve,  according  as  E'  is  greater,  equal  to,  or 
less  than  40/  u  ■    -^      "^    ^''T.   ^  ^^J^^^^r^^r^^^i  - 

233.  Conclusions.    Hence  we  arrive  at  tne  following  results 
The  equation      ax^-\-'bxy-^cy'-\-dx^ey-\-f=^ 
represents  an  ellipse,  if  ¥—^ac  be  negative^  subject  to  three  ex- 
ceptions, in  which  it  represents  respectively  a  circle,  a  point, 
and  an  imaginary  curve  (Art.  230). 

If  y^—^ac  be  positive,  the  equation  represents  an  hyperbola, 
subject  to  one  exception  when  it  represents  two  intersecting 
straight  lines  (Art.  230). 

If  Z»'— 4«jc=0,  the  equation  represents  a  parabola,  subject  to 
three  exceptions,  in  which  it  represents  respectively  two  jparal- 
lel  straight  lines,  one  straight  line,  and  an  imaginary  curve 
(Art.  232). 

Ex.  1.  Determine  the  form  and  situation  of  the  curve  repre- 
sented by  the  equation 

aj''_£c?/4-2/'-2a?-2?/+2  =  0. 

Here  ¥—^ac:=—Z\  hence  the  equation  represents  an  el- 
lipse (Art.  233). 

In  order  to  transfer  the  origin  to  the  centre  of  the  curve,  we 
substitute  h-\-x!^  for  x,  and  Tc-\-y'  for  y.  The  values  of  h  and  It 
are  given  by  the  formulas  of  Art.  229, 

,     -4-2         ^     ,     -4-2 

also,  /'=4-4-l-4-4-44-2=-2. 


176 


ANALYTICAL   GEOMETKY. 


Hence  tlie  transformed  equation  is 

[N'ext,  retaining  the  centre  of  the  ellipse  as  the  origin,  we 
must  find  through  what  angle  the  axes  must  be  turned  in  order 
that  the  term  containing  xy  may  vanish. 

By  Art.  230,  tang.  20= =^=infinity;  hence  20=90°, 

and  0=45°. 

Also,  by  Art.  231,    A=|(2  + Vl)=|, 
and  B=-i(2-Vl)=i. 

Therefore  the  equation  to  the  ellipse  referred  to  the  new  axes  is 


x^     3?/' 

2+i-2^ 


0, 


or  a;''+3y'=4. 

2  4 

The  semiaxes  are  — ^  and  2,  and  the  axes  are  —j^  and  4. 

The  annexed  figure  represents  the  form  of  the  curve,  and 
its  position  with  respect  to  the  different 
'^     systems  of  axes,  the  co-ordinates  of  A' 
being  (2, 2),  and  the  angle  X'A'X^'  be- 
,  ing  45°. 

■'  a;'— a?y+2/'-2^-2y+2  =  0 

is  the  equation  of  the  ellipse  referred 
to  the  axes  AX,  AY. 
■^  a;'-a;?/4-2/'-2=0 

is  the  equation  of  the  same  ellipse  referred  to  the  axes  A'X', 
A'Y^ 

aj'4-32/'  =  4 
is  the  equation  of  the  same  ellipse  referred  to  the  axes  AJX!', 

Ex.  2.  Determine  the  form  and  situation  of  the  curve  repre- 
sented by  the  equation 

a;'— 6i»?/+2/'-6i2?+2y+5=0. 

Here  ^'— 4ac=:36— 4=32;  hence  the  equation  re^^resents 
an  hyperbola. 

By  the  formulas  of  Art.  229  we  find 


v.     ! 


't 


GENEEAL  EQUATION  OF  THE  SECOND  DEGREE.      177 

'     -13+12  ^_iz!^_     1. 


'"-       32       — '  '"    32 

/'  =  l-2  +  5=:4. 

Hence,  when  the  origin  is  transferred  to  the  point  (0,  —  1),  the 
equation  becomes       «'— 6iC2/+y^+4=0. 

In  order  that  the  term  containing  xy  may  vanish,  we  must 

have  tang.  20=— y-= infinity.     Hence  0=45°. 

Also,  A=-|(2+V36)=+4, 

and  B=i(2--v/36)=-2. 

Hence  the  transformed  equation  is 

4y^-2«H4  =  0. 
The  student  should  construct  a  figure  showing  the  form  and 
position  of  the  curve  with  respect  to  the  different  axes  of  ref- 
erence. 

Ex.  3.  Determine  the  form  and  situation  of  the  curve  repre- 
sented by  the  equation 

aj'-2icy+2/'-8a;+16  =  0.  (1) 

Here  5"— 4<3^c=0 ;  hence  the  equation  represents  a  parabola. 
Substituting  for  x  in  eq.  (1), 

x'  COS.  %—y'  sin.  0, 
and  for  y,  x'  sin.  0+2/'  cos.  0, 

we  obtain  an  equation  of  the  form 

Aa;'+Ba^y+C2/'+DaJ4-Ey+F=0,  (2) 

where  A  =  1 — sin.  20,  D  =  —  8  cos.  0, 

B  =  -2(cos.'0-sin.'0),  E=     8  sin.  0. 

C=     l4-sin.20,  r=16. 

Now,  in  order  that  B  may  vanish,  we  must  have 

COS.  0=sin.  0 ;  that  is,  0=45°. 
Making  0=45°,  equation  (2)  becomes 

or  j/'+y.  2^/2—0;.  2  V2  +  8=0, 

which  may  be  written 

y'+y.2V2+2=aj.2V2-6, 


or 


«-  ^     ^ 


.(2/+V2)'=2V2(a?--^). 


C'f'  o     C?        -^     C-    yiz.-^  --       '-^       -^    V.  ■■' 


178  ANALYHCAli   GEOMETRY. 

If  now  we  transfer  the  origin  to  a  point  whose  co-ordinates 
are  x = — r^,  and  y=  —  ^2, 

the  equation  to  the  curve  will  become 

The  student  should  construct  a  figure  showing  the  form  and 
position  of  the  curve  with  respect  to  the  different  axes  of  ref- 
erence. 

234.  Equation  to  the  conic  sections  referred  to  the  same 
axes  and  origin.  When  the  origin  of  co-ordinates  is  placed  at 
the  vertex  of  the  major  axis,  the  equation  of  the  ellipse  (Art. 

129)  is  y-J^l^ax-x'')', 

the  equation  to  the  hyperbola  for  a  similar  position  of  the  or- 

igin  (Art.  180)  is  y'=-l2ax-^x'')\ 

the  equation  to  the  circle  (Art.  63)  is 

y^=2rx—x^', 
and  the  equation  to  the  parabola  (Art.  85)  is 

y^=^ax. 
These  equations  may  all  be  reduced  to  the  form. 

y'^=mx-\-nx^. 

In  the  ellipse,        m= — ,  and  n— — j-; 
a  a 

21/  V' 

in  the  hyperbola,      m— — ,  and  n=-ii\ 

a  a 

in  the  parabola,     m—^a^  and  7i=0. 

In  each  case  m  represents  the  latus  rectum  of  the  curve,  and 
n  the  square  of  the  ratio  of  the  semiaxes.  In  the  ellipse  n  is 
negative,  in  the  hyperbola  it  is  positive,  and  in  the  parabola  it 
is  zero. 

The  equation  y^='mx-\-nx^  is  the  simplest  form  of  the  equa- 
tion to  the  conic  sections  taken  collectively^  and  referred  to  the 
same  axes  and  origin. 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


179 


235.  Miscellaneous  Examples. 

Draw  the  curves  of  which  the  following  are  the  equations : 


Ex.1.  x^  +  27f  =  10, 
Ex.2.  x'-'^i/^lO. 
Ex.3.  x^^-Zx=:10y. 
Ex.4.  xij+10y=^0. 
Ex.6.  3aj''  +  2/  =  18. 
Ex.6.  3aj''+22/"  =  -18. 
Ex.7.  3a;'+2y'=aY^v^f 


Ex.   8.  2/'=%- 3). 
V'/f-Ex.    9.  3x7/ =6.  yf 

Ex.10.  3a?.?/-aj+2=0. 
Ex.11.  5x'+7y'=ll.'C£^ 
Ex.12.  3y''-2y+4a;=0, 
Ex.13.  ^'^  +  52/-9a;  +  10=a 
Ex 


ir-ic 


.14  W -112^'= -60.  ^^ 


180 


ANALYTICAL   GEOMETRY. 


SECTION  IX. 

LINES    OF   THE    THIRD   AND    HIGHER   ORDERS. 

236.  Lines  of  the  third  order  have  their  equations  of  the  form 

ay^ + Jy'a? + cydi^  +  dx^  +  ey^  -\-fyx  -^gx'^+hy-^lcx+l=0. 
Newton  has  shown  that  all  lines  of  the  third  order  are  com- 
prehended under  some  one  of  these  four  equations : 
j  "^  (1)  xy^-i'ey=ax^  +  bx''-{-cx-}-d; 
1  (2)  xy=ax^  +  bx'^-\-cx+d; 

6~  (3)  y''=ax^-\-bx''-{-€x+d; 

\  (4)  y=ax'  +  bx'-\-cx-{-d;        ,^^  ^i;^^ 

in  which  a,  b,  c,  d,  e  may  be  positive,  negative,  or  evanescent, 
excepting  those  cases  in  which  the  equation  would  thus  become 
one  of  an  inferior  order  of  curves. 

The  first  equation  comprehends  seventy-three  different  spe- 
cies of  curves,  the  second  only  one,  the  third  five,  and  the  fourth 
only  one,  making  eighty  different  species  of  lines  of  the  third 
order. 


237.  It  is  not  proposed  to  attempt  any  general  investigation 
of  the  equation  of  the  third  degree,  but  merely  to  select  a  few 
instances  calculated  to  exhibit  the  properties  of  some  of  the 
more  remarkable  curves. 

Ex.  1.  Trace  the  curve  whose  equation  is  6y=x^. 
S  uppose  x=0,    th  en  y = 0. 

x=±l,  "    2^=  ±0.167. 

a;=±2,  "    y=  ±1.333. 

•X  a;=±3,  "    y=  ±4.500. 

aj=±4,  "    y=  ±10.667,  etc. 
Constructing  these  values,  we  obtain  the 
figure  annexed.     This  equation  may  be 
written  more  generally  ay=x^,  and  the  curve  is  called  the  cu- 
bical j)arabola.     It  belongs  to  eq.  (4),  Art.  236. 


LINES    OF   THE   THIED    AND    HIGHER   OEDEES. 


181 


Ex.  2.  Trace  tlie  curve  whose  equation  is  4?/': 
Suppose      x—0,    then  y=0. 

x=+l,   "     y:^zt0.600. 

ic=+2,   "     ?/=  ±1.414. 

aj=+3,   "     2/=  ±2.598. 

aj=+4,   "     ?/=  ±4.000. 

a;=+5,  "  ?/=  ±5.590. 
If  X  is  negative,  2/  becomes  imaginary.  The 
curve  is  represented  by  the  annexed  figure,  and  is  called  the 
semicicbical  jfarabola.  The  equation  in  a  more  general  form 
is  a]/'=x^,  and  belongs  to  eq.  (3)  of  Art.  236. 

Ex.  3.  Trace  the  curve  whose  equation  is 
a;y=10. 

Suppose  y=0,  then  a? = infinite. 

"       'X  is  negative,  "    y  is  impossible. 
"        y=±l,  "    «=+10,etc. 

The  curve  is  of  the  form  represented  in  the 
annexed  figure,  and  belongs  to  equation  (1), 
Art.  236. 

Ex.  4.  Trace  the  curve  y=x^—x. 
Suppose     x=0,       then  y=0. 

x=±0.5,   "    2/==f0.375. 

aj=:±l,      "    y^O. 

x=±:2,     "   y=±Q' 

The  curve  is  shown  in  the  annexed  figure,  and 
belongs  to  eq.  (4),  Ai«t.  236. 

Ex.  5.  Trace  the  curve  y^=x^-'X. 
Suppose    x=0,       then  y=0. 

a;=±l,      "     y=0. 

x=-{-0.5    "    y=:  impossible. 

a;=— 0.5,   "     ?/=  ±0.612. 

a?=+2,      "    2/=  ^2.449. 

aj=+3,      ''    2/=  ±4.899. 
The  curve  is  shown  in  the  annexed  figure. 


182 


ANALYllCAL   GEOMETRY. 


Ex.  6.  Trace  the  curve  whose  equa- 
tion is 

.,    0         V-   5_    ; .  "v ,  %" 
Ex.  7.  Trace  the  curve  whose  equa- 
tion is 


Ex.  8.  Trace  the  curve  whose  equa- 
tion is 

10y'=x'-9x''  +  24:x-16. 


Ex.  9.  Trace  the  curve  whose  equation  is 
10y'=x'-12x''-{-4.8x-64:. 


Ex.  10.  Trace  the  curve  whose  equation  is 
10y^=x'-\-Sx''—22x-^2^: 


Ex.  11.  Trace  the  curve  whose  equation  is 
y=x^—Sx. 

Ex.  12.  Trace  the  curve  whose  equation  is 
^=^x'--9x. 

Ex.  13.  Trace  the  curve  whose  equation  is 
y^—x^—Qi^. 


V. 


238.  Equations  of  the  fourth  degree.  The  general  equation 
of  the  fourth  degree  represents  an  immense  variety  of  curve 
lines,  the  number  of  different  species  being  estimated  at  more 
than  5000.     The  number  of  species  of  lines  of  the  fifth  and 


LINES   OF   THE   THIRD   AND   HIGHER   OEDEKS.  183 


higher  orders  is  so  great  as  to  preclude  any  attempt  to  enumer- 
ate them  completely. 


Ex.  1.  Trace  the  curve  whose  equation  is 


Ex.  2.  Trace  the  curve  whose  equa- 
tion is 

27y=x*-20x'+64:, 


Ex.  3.  Trace  the  curve  whose  equation  is 


Ex.  4.  Trace  the  curve  whose  equation  is 
X* + 2i»y + 2/* = («' + yj = 16(ic'  -  y'). 


184 


ANALYTICAL    GEOJkLETRY. 


SECTION  X. 


TRANSCENDENTAL   CURVES. 


239.  Equations  classified.  Equations  may  be  divided  into 
two  classes,  algebraic  and  transcendental.  An  algebraic  equa- 
tion between  two  variables,  x  and  y,  is  one  which  can  be  reduced 
to  a  finite  number  of  terms  involving  only  integral  powers  of 
X  and  y,  and  constant  quantities.  Equations  which  can  not  be 
thus  reduced  are  called  transcendental ;  for  they  can  only  be 
expanded  into  an  infinite  series  of  terms,  in  w^hich  the  power 
of  the  variable  increases  without  limit,  and  the  equation  tran- 
scends all  finite  orders. 

240.  Curves  classified.  Curves  whose  equations  are  'tran- 
scendental are  called  transcendental  curves.  Among  tran- 
scendental curves,  the  cycloid  and  the  logarithmic  curves  are 
the  most  important.  The  logarithmic  curve  is  useful  in  exhib- 
iting the  law  of  the  diminution  of  the  density  of  the  atmos- 
phere, and  the  cycloid  in  investigating  the  laws  of  the  pendu- 
lum and  the  descent  of  heavy  bodies  toward  the  centre  of  the 
earth. 


CYCLOID. 

241.  A  cycloid  is  the  curve  described  by  a. point  in  the  cir- 
cumference of  a  circle  rolling  in  a  straight  line  on  a  plane. 


A.         N  D 

Thus,  if  the  circle  EPN  be  made  to  roll  in  a  given  plane 
upon  a  (straight  line  AC,  the  point  P  of  the  circumference, 


TEANSCENDENTAL   CUEVES. 


185 


wliicli  was  in  contact  with  A  at  the  commencement  of  the  mo- 
tion, will  in  a  revolution  of  the  circle  describe  a  curve  ABC, 
which  is  called  the  cycloid.  The  circle  EPJST  is  called  the  gen- 
erating circle,  and  P  the  generating  point. 

When  the  point  P  has  arrived  at  C,  having  described  the  arc 
ABC,  if  it  continue  to  move  on,  it  will  describe  a  second  arc 
similar  to  the  first,  and  so  on  indefinitely.  As,  however,  in  each 
revolution  of  the  generating  circle  an  equal  carve  is  described, 
it  is  only  necessary  to  examine  the  curve  ABC,  described  in 
one  revolution  of  the  o-eneratino^  circle. 


242.  After  the  circle  has  made  one  revolution,  every  point 
of  the  circumference  will  have  been  in  contact  with  AC,  and 
the  generating  point  will  have  arrived  at  C.  The  line  AC  is 
called  the  base  of  the  cycloid,  and  is  equal  to  the  circumfer- 
ence of  the  generating  circle.  The  line  BD,  drawn  perpen- 
dicular to  the  base  at  its  middle  point,  is  called  the  axis  of  the 
cycloid,  and  is  equal  to  the  diameter  of  the  generating  circle. 

243.  To  find  the  equation  of  the  cycloid.  Let  us  assume  the 
point  A  as  the  origin 
of  co-ordinates,  and 
let  us  suppose  that 
the  generating  point 
has  described  the  arc 
AP.  If  N  designates 
the  point  at  which  the  generating  circle  touches  the  base,  it  is 
plain  that  the  line  AN  will  be  equal  to  the  arc  PK  Through 
N  draw  the  diameter  EN,  which  will  be  perpendicular  to  the 
base  of  the  cycloid.  Through  P  draw  PII  parallel  to  the 
base,  and  PR  perpendicular  to  it.  Then  PR  will  be  equal  to 
UN",  which  is  the  versed  sine  of  the  arc  PIST. 

Let  AR=:a?,  and  PR  or  irN"=?//  and  let  t  represent  the  ra- 
dius of  the  generating  circle.    By  Geom.,  Bk.  iy.,Prop.  23, Cor., 

RN=PH=VNHxHE  =  V2/(2r-2/)  =  V2/'2/-?/''; 
also,  AR=AK-\RN=arc  PN-PH. 


186  ANALYTICAL   GEOMETEY. 

The  arc  PN  is  the  arc  whose  versed  sine  is  UN  or  y. 
Substituting  the  values  of  AR,  AN,  and  RK,  we  have 
a?r=:(the  arc  whose  versed  sine  is  y)—'V'2iry—y\ 
which  is  the  equation  of  the  cycloid. 

2^.^.  Another  form  of  the  equation.  It  is  soniethnes  con- 
venient, in  the  equation  of  the  cycloid,  to  employ  the  angle  of 
rotation  of  the  generating  circle,  or  the  angle  subtended  by  the 
arc  PN  at  the  centre  of  the  circle  EPN.  Let  this  angle  be 
denoted  by  0,  and  the  radius  of  the  circle  by  r ;  then 

the  arc  PN=r0, 
and  AR  or  x=r9—r  sin.  6, 

and  HlSr  or  y=r—r  cos.  6. 

If  we  eliminate  6  from  these  two  equations,  we  shall  obtain 
the  same  value  of  x  as  given  in  Art.  243. 

LOGABITHMIC  CURVE. 

246.  The  logarithmic  curve  takes  its  name  from  the  prop- 
erty that,  when  referred  to  rectangular  axes,  any  abscissa  is 
equal  to  the  logarithm  of  the  corresponding  ordinate.  The 
equation  of  the  curve  is  therefore 

a?=log.  y. 

If  a  represent  the  base  of  a  system  of  logarithms,  we  shall 
have  (Alg.,  Art.  394)  y=a^ 

To  examine  the  course  of  the  curve,  we  find,  when  x=0, 
yz=a!^=l'^  as  x  increases  from  0  to  oo,  ?/  increases  from  1  to  qo  ; 
as  —  ic  increases  to  oo,  y  decreases  from  1  to  0.  Draw  AB  per- 
pendicular to  DC,  and  make  it  equal  to  the  linear  unit ;  then 
the  curve  proceeding  from  B  to  the  right  of  AB  recedes  fi'om 
the  axis  of  x,  and  on  the  left  continually  approaches  that  axis, 
which  is  therefore  an  asymptote. 

Any  number  of  points  of  the  curve  may  be  determined  from 
the  equation  y=a'.  Let  AC  be  divided  into  portions  each 
equal  to  AB.  Let  a  be  taken  equal  to  the  base  of  the  given 
system  of  logarithms,  for  example  1.6,  and  let  a",  a\  etc.,  cor- 


TKANSCENDENTAL   CURVES. 


187 


respond  in  length  with  the 
different  powers  of  a.  Then 
the  distances  from  A  to  1,  2, 
3,  etc.,  will  represent  the  loga- 
rithms of  «,  «^,  a^^  etc. 

The  logarithms  of  numbers 
less  than  a  unit  are  negative^ 
and  these  are  represented  by 
portions  of  the  line  AD  to  the 
left  of  the  origin. 


246.  In  a  similar  manner  we  may  construct  the  curve  for 
any  system  of  logarithms.     Thus,  for  the  l!Taperian  system, 

«=  2.718. 
o?=  7.389. 
a' =20.085. 
ar'^  0.368. 
«-'=  0.135,  etc. 
If  at  the  point  A  we  erect  an 
ordinate  equal  to  unity,  at  the 
point  1  an  .  ordinate   equal  to 
2.718,  at  the  point  2  an  ordinate 
equal  to  7.389,  etc.,  at  the  point  —1  an  ordinate  equal  to  0.368, 
etc.,  the  curve  passing  through  the  extremities  of  these  ordi- 
nates  will  be  the  logarithmic  curve  for  the  Naperian  base. 

Ex.  1.  Construct  by  points  the  logarithmic  curve,  tlie  base 
being  10. 

Ex.  2.  Construct  by  points  the  logarithmic  curve,  the  base 
being  \. 

CURVE  OF  SINES,  TANGENTS,  ETC. 

247.  If  we  conceive  the  circumference  of  a  circle  to  be  ex- 
tended out  in  a  right  line,  and  at  each  point  of  this  line  a  per- 
pendicular ordinate  to  be  erected  equal  to  the  sine  of  the  cor- 
responding arc,  the  curve  line  drawn  through  the  extremity  of 
each  of  these  ordinates  is  called  the  curve  of  sines. 


188 


ANALYTICAL   GEOMETRY. 


Draw  a  straight  line  ABC  equal  to  tlie  circumference  of  a 

given  circle,  and 
upon  it  lay  off  the 
lengths  of  several 
arcs,  at  every  10°  for 
example,  from  0°  at 
A  to  360°  at  C ;  from  these  points  draw  perpendicular  ordi- 
nates  equal  to  the  sines  of  the  corresponding  arcs,  upward  or 
downward,  according  as  the  sine  is  positive  or  negative  in  that 
part  of  the  circle ;  then  draw  a  curve  line  ADBEC  through  the 
extremities  of  all  these  ordinates ;  it  will  be  the  curve  of  sines. 

248.  To  find  the  equation  of  the  curve  of  sines.    Draw  any 
ordinate  PM.    Let  AM=a?,  and  PM=2//  then  the  equation  is 

?/=sin.  a;. 
If  T  represent  the  radius  of  the  given  circle,  then 


y—T  sm.  -. 

Since  the  sine  is  0  when  the  arc  is  0,  the  curve  cuts  the  axis 
at  A.  Since  the  sine  of  90°  is  a  maximum,  the  highest  point 
of  the  curve  will  be  at  D,  where  y=r.  The  curve  cuts  the 
axis  again  in  B ;  from  B,  y  increases  negatively  until  it  equals 
— r,  and  then  decreases  to  0,  so  that  we  have  a  second  branch 
equal  and  similar  to  the  first.  Beyond  C  the  values  of  y  recur, 
and  the  curve  continues  the  same  course  ad  infinitum.  Also, 
since  sin.  (— ic)=  —sin.  a?,  there  is  a  similar  branch  to  the  left 
of  A. 

In  a  similar  manner  may  be  drawn  the  curve  of  tangents, 
the  curve  of  secants,  etc. 

SPIRALS. 
249.  Definition.  If  a  right  line  be  revolved  uniformly  in 
the  same  plane  about  one  of  its  points  as  a  centre,  and  if  at  the 
same  time  a  second  point  travel  along  the  line  in  accordance 
with  some  prescribed  law,  the  latter  point  will  generate  a  curve 
called  a  spiral. 


TBANSCENDENTAL    CUEVES. 


189 


Thus,  let  PD  be  a  straight  line  which  revolves  uniformly 
round  the  point  P,  starting  from 
the  position  PC,  and  at  the  same 
time  let  a  point  move  from  P 
along  the  line  PD  according  to 
some  prescribed  law ;  the  point 
will  trace  out  a  curve  line  which 
commences  at  P,  and  after  one 
revolution  will  arrive  at  a  point 
A ;  after  two  revolutions  it  will 
arrive  at  a  point  B,  and  so  on. 
The  curve  thus  traced  is  called  a  spiral. 


250.  The  fixed  point  P,  about  which  the  right  line  revolves, 
is  called  the  pole  of  the  spiral.  The  portion  of  the  spiral  gen- 
erated while  the  straight  line  makes  one  revolution  is  called  a 
spire.  If  the  revolutions  of  the  radius  vector  are  continued 
indefinitely,  the  generating  point  will  describe  an  unlimited 
spiral.  It  is  assumed  that  the  point  does  not,  after  a  limited 
number  of  revolutions,  describe  again  the  previous  curve,  but 
that  any  straight  line  drawn  through  the  pole  of  the  spiral  will 
cut  the  curve  in  an  infinite  number  of  points. 

Instead  of  starting  from  the  pole,  the  generating  point  may 
commence  its  motion  at  any  distance  from  the  pole;  and  in- 
stead of  receding,  it  may  move  toward  the  pole. 

With  P  as  a  centre,  and  any  convenient  radius  as  PA,  de- 
scribe the   circumference  ADE ; 
the  angular  motion  of  the  radius 
Yector   about   the   pole   may  be 
measured  by  the  arcs  of  this  cir-      / 
cle,  estimated  from  A.     It  is  gen-  e: 
erally  convenient  to  make  the  ra-     \ 
dius  of  the  measuring  circle  equal 
to  the  length  of  the  radius  vector 
at  the  end  of  one  revolution  of 


190 


ANALYTICAL   GEOMETEY. 


the  generating  point,  starting  from  the  pole,  but  the  measuring 
circle  may  have  any  magnitude. 

251.  Spiral  of  Archimedes.  While  the  line  PD  revolves 
uniformly  round  the  point  P,  let  the  generating  point  also 
move  uniformly  along  the  line  PD ;  it  will  describe  the  spiral 
of  Archimedes. 

252.  To  construct  the  spiral  of  Archhneaes.  Let  P  be  the 
pole,  and  PX  the  first  position  of  the 
radius  vector.  With  P  as  a  centre, 
and  any  convenient  radius,  describe 
the  measuring  circle  ACEG,  and  di- 
vide its  circumference  into  any  con- 
venient number  of  equal  parts,  as,  for 

example,  eight.  On  PB  set  off  PI  any  convenient  distance ; 
on  PC  set  off  PKr=2PI;  on  PD  set  off  PL=.3PI,  etc  The 
curve  passing  through  the  points  I,  K,  L,  M,  etc.,  thus  deter- 
mined, will  be  the  spiral  of  Archimedes,  for  the  radii  vectores 
are  proportional  to  the  arcs  AB,  AC,  etc.,  of  the  measuring 
circle. 


263.  To  find  the  equation  to  the  spiral  of  Archimedes.  From 
the  definition  of  the  curve,  the  radii  vectores  and  the  measur- 
ing arcs  increase  uniformly ;  that  is,  in  the  same  ratio.  Hence 
we  have 

PL :  PR : :  angle  APD  :  four  right  angles. 
Designate  the  radius  vector  PL  by  r,  PR  by  h,  and  the  variable 
angle  by  0 ;  then  we  shall  have 

r\h\\B\'2Tr\ 

be  .  ^         ,         ,  . 

whence  r=z^  ;  or,  putting  a=^,  we  nave  the  equation 

r  =  ae. 
Wlien  the  radius  vector  has  made  two  revolutions,  or  0=4:tt, 
we  have  r=25;  that  is,  the  curve  cuts  the  axis  PX  at  a  dis- 
tance equal  to  2PR;  after  three  revolutions  it  cuts  the  axis 


TRANSCENDENTAL   CUEVES. 


191 


PX  at  a  distance  equal  to  3PR,  etc.  Hence  the  distance  be- 
tween any  two  consecutive  spires,  measured  on  a  radius  vector, 
is  always  the  same. 

254.  Ilyjper'holiG  sjpiral.  While  the  line  PN  revolves  uni- 
formly about  P,  let  the  generating  point  move  along  the  line 
PK  in  such  a  manner  that  the  radius  vector  shall  be  inversely 
proportional  to  the  corresponding  angle ;  it  will  describe  the 
hyperbolic  spiral. 

255,  To  find  the  equation  to  the  hyjoerbolio  spiral.    From 
the  definition  of  the  curve,  the 
radius  vector  is  inversely  propor- 
tional to  the  measuring  angle; 
hence  we  have 

PG :  PK : :  angle  APK :  four 
right  angles. 
Designate  the  radius  vector  PN 
by  r,  PG  by  h,  and  the  variable 
angle   measured  from  the  line 
PX  by  0,  and  we  shall  have 

b:r::e:27r. 
AVlience  r9=2h7r; 

or,  putting  2b7r=a,\Yeha.YG 

rO  =  a. 

When  0=0,  r=co;  as  9  increases,  7'  decreases,  at  first  very 
rapidly,  but  afterwards  more  uniformly.  As  0  may  increase 
without  limit,  r  may  decrease  indefinitely  without  actually  be- 
coming zero ;  hence,  as  the  radius  vector  revolves,  the  curve 
continues  to  approach  the  pole,  but  reaches  it  only  after  an  in- 
finite number  of  revolutions.  This  curve  is  called  the  hyper- 
bolic spiral  from  the  similarity  of  its  equation  to  that  of  the 
hyperbola  referred  to  its  asymptotes  {xy=c'),  the  product  of 
the  variables  r  and  6  being  equal  to  a  constant  quantity. 


192 


ANALYTICAL   GEOMETEY. 


256.  To  construct  the  hyjperbolic  spiral.  Let  P  be  the  pole, 
and  PX  the  first  position  of  the  radius  vector.  With  any  con- 
venient radius  draw  the  measuring  circle  ABDE,  and  divide 
its  circumference  into  any  convenient  number  of  equal  parts 
AB,  BC,  CD,  etc.  On  PB,  produced  if  necessary,  take  any  con- 
venient distance,  as  P^N" ;  take  PM  equal  to  one  half  of  PN, 
PL  equal  to  one  third  of  PJ^,  PK  equal  to  one  fourth  of  PN, 
etc. ;  the  curve  passing  through  the  points  N,  M,  L,  K,  etc.,  will 
be  an  hyperbolic  spiral,  because  the  radii  vectores  are  inverse- 
ly proportional  to  the  corre!^ponding  angles  measured  from 
PX. 

257.  Logarithmic  spiral.  While  the  line  PA  revolves  uni- 
formly about  P,  let  the  generating  point  move  along  PA  in 
such  a  manner  that  the  variable  angle  may  be  proportional  to 
the  logarithm  of  the  radius  vector;  it  will  describe  the  loga- 
rithmic spiral. 

The  equation  of  the  logarithmic  spiral  is 

*=  a' 

or  r=ah\ 

h  being  the  base  of  the  system  of  logarithms  (Alg,,  Art.  394), 
and  a  any  arbitrary  constant. 


258.  2'o  construct  the  logarithmic  spiral.    If  we  take  5  =  10, 

the  base  of  the  common  system  of 
logarithms,  the  changes  of  r  are  so 
rapid  that  we  can  represent  only  a 
small  arc  of  tlie  curve.  We  will 
therefore  assume  h  =  1.2.  When 
0  —  0,  r^a,  which  determines  the 
point  L.  When  0  =  1,  that  is,  57°.3 
(radius  being  unity),  r = 1.2fl^,which 
determines  the  point  M.  When 
d  —  2^  that  is,  114°.G,  r=l.'2i^a,  or  lA4:a,  which  determines  the 


TRANSCENDENTAL   CUEVES.  193 

point  N",  etc.    As  0  increases,  r  also  increases,  but  does  not  be- 
come infinite  until  9  becomes  infinite. 

If  we  suppose  the  radius  yector  to  revolve  in  the  negative 
direction  from  PA,wlien  9=— 1,7^=0. 83a,  v^hidi  determines, 
another  point  of  the  curve.  When  6  =—2,  r=0.Q9a,  etc. 
Hence  we  see  that,  as  the  radius  vector  revolves  in  the  nega- 
tive direction,  it  generates  a  portion  of  the  spiral  which  slowly 
approaches  the  pole, but  can  not  reach  it  until  0=—oo. 

Thus  we  see  that  the  logarithmic  spiral  makes  an  infinite 
series  of  convolutions  around  the  pole  P. 

I 


194  ANALYTICAL   GEOMETRY. 


PART   III. 
GEOMETRY  OF  THREE  DIMENSIONS. 

SECTION  I. 
OF    POINTS    IN    SPACE. 

259.  Hitherto  we  have  considered  only  points  and  lines  sit- 
uated in  one  plane,  and  we  have  seen  that  the  position  of  a 
point  in  a  plane  may  be  denoted  by  its  distances  from  two  as- 
sumed fixed  lines  or  axes  situated  in  that  plane.  We  have  now 
to  consider  how  the  position  of  any  point  in  space  may  be  rep- 
resented. 

260.  To  determine  the  position  of  a  point  in  space.  Let 
three  planes  XAY,  ZAX,  ZAY,  supposed  to  be  of  indefinite  ex- 
tent, be  drawn  perpendicular  to  each  other,  and  let  these  planes 
intersect  each  other  in  the  three  straight  lines  AX,  AY,  AZ. 
Let  P  be  any  point  in  space  whose  position  it  is  required  to 
determine. 

From  the  point  P  draw  the  line  PB  perpendicular  to  the 
plane  XAY ;  draw  PC  perpendicular  to 
the  plane  ZAX,  and  PD  perpendicular  to 
the  plane  ZAY;  then  the  position  of  the 
point  P  is  completely  determined  when 
_x  these  three  perpendiculars  are  known. 
Let  a,  ^,  c  represent  these  three  perpen- 


/§  B  diculars.     On  AX  take  AE=:t?,  on  AY 

^  take  AF  —  h,  and  on  AZ  take  AG = c,  and 

through  the  points  E,  F,  and  G  let  planes  be  drawn  parallel  to 
the  three  planes  ZAY,  ZAX,  and  XAY,  forming  the  rectangu- 
lar parallelepiped  EFG. 

Since  the  plane  drawn  through  E  is  every  where  distant  from 
the  plane  ZAY  by  a  quantity  equal  to  «,  the  point  P  must  be 


OF   POINTS    IN    SPACE.  195 

somewhere  in  this  plane  ;  and  since  the  plane  drawn  through 
F  is  every  where  distant  from  the  plane  ZAX  by  a  quantity 
equal  to  J,  the  point  P  must  be  also  in  this  plane.  It  must 
therefore  be  in  the  line  BP,  which  is  the  common  section  of 
these  two  planes.  Also,  since  the  plane  drawn  through  G  is 
every  where  distant  from  the  plane  XAY  by  a  quantity  equal 
to  c,  the  point  P  must  be  somewhere  in  this  plane ;  it  must 
therefore  be  at  the  intersection  of  this  third  plane  with  the 
line  BP.  Thus  the  position  of  the  point  P  is  completely  de- 
termined. 

261.  Definitions.  The  three  planes  XAY,  ZAX,  ZAY,  by 
reference  to  which  the  position  of  the  point  P  has  been  deter- 
mined, are  called  the  co-ordinate  planes.  The  first  is  desig- 
nated as  the  plane  XY,  the  second  as  the  plane  XZ,  and  the 
third  as  the  plane  YZ.  The  lines  AX,  AY,  AZ,  which  are  the 
intersections  of  these  three  planes,  are  called  the  co-ordinate 
axes.  The  first  is  called  the  axis  of  X,  and  distances  parallel 
to  it  are  denoted  by  x ;  the  second  is  the  axis  of  Y,  and  dis- 
tances parallel  to  it  are  denoted  by  y ;  the  third  is  the  axis  of 
Z,  and  distances  parallel  to  it  are  denoted  by  z.  The  point  A, 
in  w^hicli  the  three  axes  intersect,  is  called  the  origin  of  co-or- 
dinates. The  equations  of  a  point  in  space  are  therefore  of 
the  form  x—a^    2/=^?    z=c. 

262.  Signs  of  the  co-ordinates.  If  the  three  co-ordinate 
planes  be  indefinitely  produced,  there  will 
be  formed  about  the  point  A  eight  solid  an- 
gles, four  above  the  horizontal  plane  XAY, 
and  four  below  it.  It  is  required  to  denote  x 
analytically  in  which  of  these  angles  the 
proposed  point  is  situated.  For  this  pur- 
pose, if  w^e  regard  distances  measured  on 
AX  to  the  right  of  A  as  jpositive,  we  must  regard  distances 
measured  to  the  left  of  A  as  negative.  So,  also,  y  is  regarded 
as  positive  when  it  is  \n  front  of  the  plane  ZX,  and  negative 


u 

a 

ii 

ZXAY'. 

a 

a 

a 

ZX'AY'. 

ii 

i( 

ii 

ZX'AY. 

a 

a 

ii 

Z'XAY. 

a 

a 

a 

Z^XAY'. 

a 

6i 

a 

Z'X'AY'. 

a 

a 

ii 

Z'X'AY. 

196  ANALYTICAL   GEOMETEY. 

when  it  is  'behind  that  plane;  and  z  is  regarded  as  positive 
when  it  is  cibove  the  plane  XY,  and  negative  when  it  is  below 
that  plane.     Hence  the  equations  of  a  point  in  each  of  these 
eight  angles  are  as  follows : 
If  aj=  +c^,  y=  +  J,  s=  +(?,  the  point  is  in  the  angle  ZXAY. 

x-^-a,y-—b,z='\-c, 

x=—a,y=-^b,z:=-\-c, 

x=—a,y='\-b,z=-{-c, 

x=  -{-a,  y—  ■\-b,  z—  —c, 

x=  +a,  y=  —b,  z=—c, 

x=  —a,  y—  —J,  z—  —c, 

x——a^  y—-\-b^  z=—c, 

263.  Co-ordinates  of  particular  joints.  If  the  point  P  be 
situated  in  the  plane  of  xy,  then  its  distance  z  from  this  plane 
is  0,  and  its  equations  will  be 

x=±a,    y=z±b,    zt=iO. 
If  the  point  be  situated  in  the  plane  of  xz,  then  its  distance 
y  from  this  plane  is  0,  and  its  equations  will  be 
x=±a,    y=0,    z=dzc. 
If  the  point  be  situated  in  the  plane  of  yz,  then  its  distance 
X  from  this  plane  is  0,  and  its  equations  will  be 

X  —  0,      y=zdzb,      Z=dzC. 

If  the  point  be  situated  on  the  axis  of  Xy  that  is,  on  the  inter- 
section of  the  planes  xy  and  xz,  then  its  distance  from  each  of 
these  planes  is  0,  and  its  position  will  be  expressed  by  the  equa- 
tions x=dta,    y=Oj    z=0. 
So,  also,  if  the  point  be  situated  on  the  axis  of  y,  we  shall  have 

x=0,     y=:hb,    z=0; 
and  if  it  be  situated  on  the  axis  of  Zy  we  shall  have 
x=zO,    y=0,    z=ztc. 
If  the  point  be  at  the  origin,  its  position  will  be  denoted  by 
the  equations  x=Oy    y=0,    z=^0. 

Ex.  1.  Indicate  by  a  figure  the  position  of  the  point  whose 
equations  are 

x=+4:,    2/== -3,     z=-2. 


OF   POINTS   IN   SPACE.  197 

Ex.  2.  Indkate  by  a  figure  the  position  of  the  point  whose 
equations  are        3?=— 2,     y=-[-7,     2=+ 5. 

Ex.  3.  Draw  a  triangle,  the  co-ordinates  of  whose  angular 
points  are  a?=  +  3,     2/= +4,    b=+2;    .1       , 

i»=:_3,       2/=:~4,       Z^--'l\()(AVlA^    — f  - 

264.  Projections.  If  a  perpendicular  be  let  fall  from  any 
point  P  upon  a  given  plane,  the  point  in  which  this  line  meets 
the  plane  is  called  the  projection  of  the  point  P  on  the  plane. 
The  projections  of  the  point  P  (Art.  260)  on  the  three  co-ordi- 
nate planes  are  the  points  B,  C,  D. 

The  projection  of  any  curve  upon  a  given  plane  is  the  curve 
formed  by  projecting  all  of  its  points  upon  that  plane.  When 
the  curve  projected  is  a  straight  line,  its  projection  on  any  one 
of  the  co-ordinate  planes  will  also  be  a  straight  line,  for  all  the 
points  of  the  given  line  are  comprised  in  the  plane  passing 
through  this  line  and  drawn  perpendicular  to  the  co-ordinate 
plane ;  and  since  the  common  section  of  any  two  planes  is  a 
straight  line,  the  projections  of  the  points  must  all  lie  in  one 
straight  line.  This  plane,  which  contains  all  the  perpendicu- 
lars drawn  from  different  points  of  the  straight  line,  is  called 
i\iQ  2>rojecting  plane. 

If  the  positions  of  any  two  projections  of  the  point  P  are 
given,  it  will  be  sufficient  to  determine  the  point  P ;  for  a  line 
drawn  from  either  projection,  perpendicular  to  the  plane  in 
which  it  is,  necessarily  passes  through  the  point  P,  so  that  P 
will  be  at  the  intersection  of  two  such  perpendiculars.  When 
two  projections  of  a  point  are  known,  we  can  always  determine 
the  third. 

265.  To  find  the  distance  of  any  point 
from  the  origin  in  terms  of  the  co-ordi- 
nates of  that  point.  Let  AX,  AY,  AZ  be 
the  rectangular  axes,  and  P  the  given  point. 
Let  the  co-ordinates  of  P  be  AE=a?,  BE=:?/, 
andPB=2. 


B 


198  AITALYTICAL   GEOMETRY. 

The  square  on  AP=the  sum  of  the  squares  on  AB  and  PB. 

Also,  the  square  on  AB  =  the  sum  of  the  squares  on  AE 
and  EB ;  that  is,     AP^ = AE' + EB^  +  PB^ 
or  A'F'=x'-\-y'-{-3\ 

Ex.  1.  Determine  the  distance  from  the  origin  to  the  point 
whose  co-ordinates  are 

Ex.  2.  Determine  the  distance  from  the  origin  to  the  point 
whose  co-ordinates  are 

x=—h,     yzzz—^h,    z  =  Sh. 

266.  To  find  the  distance  between  two  given  joints  in  space. 
Let  M  and  N  be  the  two  given  points,  their  co- 
iq-  ordinates  being  respectively  x,  y,  ^,  and  x\  y\  z\ 
R  If  the  points  M  and  K  be  projected  on  the 
~^  plane  of  xy^  the  co-ordinates  a?,  y  of  the  projec- 
tions m  and  n  will  be  the  same  as  those  of  the 
'  ^  "'  '^  points  M  and  N.  f lence,  for  the  distance  mn 
we  liave  (Art.  21) 

mn'^{x-xy-^{y-y')\ 
ISTow,  if  MR  be  drawn  parallel  to  mn^  MRi^  will  be  a  right 
angle,  and  hence  MN' = MR^  -f  NR' 

=MR'+(]^^-Rn)'; 
that  is,  MN"  =  'y/{x-xy-\-{y-yy-\-{z-zy ; 

that  is,  the  distance  between  any  two  given  points  is  the  diag- 
onal of  a  right  parallelopiped,  whose  three  adjacent  edges  are 
the  differences  of  the  parallel  co-ordinates.  , 

Ex.1.  Determine  the  distance  between  the  points        ||       \ 
x=Z,    y=4:,        ands=— 2,  Vyi«' 

and  x=4:,    y=z—3,    ands— 1.  Ane.  y^9. 

Ex.  2.  Determine  the  distance  between  the  points 
x=2,        y=2,        0=1, 
and  x=-~2,    y=-^B,    z—^.  An^. 


THE   STRAIGHT  LINE   IN   SPACE. 


199 


SECTION  II. 

THE   STRAIGHT  LINE  IN   SPACE. 

267.  A  straight  line  may  be  regarded  as  tlie  common  section 
of  two  planes,  and  therefore  its  position  will  be  known  when 
the  position  of  these  planes  is  known ;  hence  its  position  may 
be  determined  by  the  projecting  planes,  and  the  situation  of 
the  projecting  planes  is  given  by  their  intersections  with  the 
co-ordinate  planes;  that  is, by  the  projections  of  the  given  line 
upon  the  Co-ordinate  planes. 


268.  To  find  the  equation  of  a  straight  line  in  sjpace. 

Let  x=mz-\-a 

be  the  equation  of  a  straight  line  M^' 
in  the  plane  of  xz,  and  through  this  line 
let  a  plane  be  drawn  perpendicular  to 
the  plane  X3.  Also,  let 
y=nZ'{-b 
be  the  equation  of  a  line  mj?  in  the 
plane  of  yz,  and  through  this  line  let  a 
plane  be  drawn  perpendicular  to  the  plane  yz.  These  two 
planes  will  intersect  in  a  line  MP,  which  will  thus  be  com- 
pletely determined.     The  two  equations 

x=mz-\-a,  (1) 

y-nz^-l^  (2) 

taken  together,  may  therefore  be  regarded  as  the  equations  of 
the  line  MP,  and  from  these  equations  the  line  MP  may  be 
constructed ;  for,  if  a  particular  value  be  assigned  to  either  va- 
riable in  these  equations,  the  values  of  the  other  two  variables 
can  be  found,  and  these  three  quantities  taken  together  will  be 
the  co-ordinates  of  a  point  of  the  required  line. 

Thus,  suppose  n'r  to  be  a  value  of  z;  this,  with  the  corre- 
sponding value  of  X  deduced  from  equation  (1),  will  determine 


200  ANALYTICAL   GEOMETET. 

a  point  n\  through  which  a  line  miiat  be  drawn  perpendicular 
to  the  plane  xz.  The  same  value  of  ^,  with  the  corresponding 
value  of  y  deduced  from  equation  (2),  will  determine  a  point 
n^  through  which  if  l^n  be  drawn  perpendicular  to  the  plane 
yz^  it  will  intersect  the  line  '^n' ^  since  both  lines  are  situated  in 
tlie  same  plane,  viz.,  a  plane  parallel  to  xy^  and  at  a  distance 
from  it  equal  to  z.  The  point  N  of  the  line  MP  is  therefore 
determined,  and  in  the  same  manner  we  may  determine  any 
number  of  points  of  this  line.  Hence  the  equations  to  the 
straight  line  MP  are         x—mz-\-a^  (1) 

y^nz-^-h.  (2) 

269.  InteTpretation  of  the  constants  in  these  equations.  In 
equation  (1)  m  represents  the  tangent  of  the  angle  which  the 
projection  of  the  given  line  on  the  plane  xz  makes  witli  the 
axis  of  2,  and  a  represents  the  distance  cut  from  the  axis  of  X 
by  the  same  projection  (Art.  29). 

In  equation  (2)  n  represents  the  tangent  of  the  angle  which 
the  projection  on  the  plane  yz  makes  with  the  axis  of  2,  and  h 
is  the  distance  cut  from  the  axis  of  Y. 

If  we  combine  these  two  equations,  and  eliminate  the  varia- 
ble z^  we  shall  have  n 

which  expresses  the  relation  between  the  co-ordinates  of  the 
point  R,  which  is  the  projection  of  the  point  N  on  the  plane 
xy^  and  therefore  this  last  equation  is  the  equation  of  the  line 
MP  projected  on  the  plane  xy. 

Ex.  The  equations  of  the  projections  of  a  straight  line  on  the 
co-ordinate  planes  zx^  zy  are 

aj=22  +  3,    2/=3^— 5; 
required  its  equation  on  the  plane  xy.       Ans.  2y=3ic— 19. 

270.  To  determine  tJie  jpoints  where  the  co-ordinate  jplanes 
are  pierced  hy  a  given  straight  line.  At  the  point  where  a 
line  pierces  the  plane  a??/  the  value  of  z  must  be  0.     If  we  sub- 


THE    STJRAIGHT   LINE   IN    SPACE.  201 

stitute  this  value  of  z  in  equations  (1)  and  (2)  of  Art.  268,  we 
shall  find  x=a^    V—^y 

hence  a  and  h  taken  together  are  the  co-ordinates  of  the  point 
in  which  the  given  line  pierces  the  plane  xij. 

In  like  manner,  the  co-ordinates  of  the  point  in  which  the 
line  pierces  the  plane  xz  may  be  determined  by  putting  2/=0 
in  equation  (2),  and  substituting  the  resulting  expression  for  z 
in  equation  (1).  In  the  same  manner,  the  point  where  the  line 
pierces  the  plane  yz  may  be  determined. 

Ex.  1.  Determine  the  points  where  the  co-ordinate  planes  are 
pierced  by  the  line  whose  equations  are 

aj  =  254-3, 
y=.Zz-n. 

Ex.  2.  Determine  the  points  where  the  co-ordinate  planes  are 
pierced  by  the  line  whose  equations  are 

^.^-2^-5, 

271.  To  find  the  equations  of  a  straight  Ime  passing  through 
a  given  jpoint.  Let  the  co-ordinates  of  the  given  point  be  x\ 
y\  z\  and  let  the  equations  to  the  straight  line  be 

x=7nz-\-aj     y=nz-{-h. 
Now,  since  this  line  passes  through  the  given  point,  we  must 
have  x'  =  mz'  +  a^ 

y'=nz'  +  h; 
hence  we  obtain 

x—x' —m{z—z\ 
and  y—y'z=7i{z—z'), 

which  are  the  equations  sought,  and  characterize  every  straight 
line  which  can  be  drawn  through  the  point  x\  y',  z'.     If  tlie 
given  point  be  the  origin,  then  x'  =  0, 2/'  =  ^?  a^d  s'  =  0,  and  the 
equations  of  a  line  passing  through  the  origin  are 
x=7nZy     y—nz. 

272.  Equations  of  a  straight  line  jpassing  through  two  given 
points.    Let  the  co-ordinates  of  the  given  points  be  x\  y' ^  z\ 

12 


202  ANALYTICAL    GEOMETRY. 

and  x'\  y'\  z" ;  then  the  equations  of  the  line  passing  through 
the  first  of  these  points  are 

x-x'^m{z-z'\\ 

y-y'=n{z-z').]  W 

Since  the  line  passes  through  the  point  x"^  y'\  z'\  we  must  also 
have  x" —x' =m{z" —z'), 

and         ^  ^y-^y'^n{z"-z'), 

from  which  we  obtain  the  values  of  m  and  n,  viz. : 
x"-x'  y"-y' 

z  —z^  z  —z 

These  values  of  m  and  n,  being  substituted  in  equation  (1), 
will  furnish  the  equations  of  the  line  passing  through  both  the 
given  points.     We  have,  therefore, 

If  one  of  the  points  x",  y"^  z"  be  the  origin,  these  equations 

become  x^  —  .z. 

z      ' 

y' 

Ex.  1.  Find  the  equations  to  the  straight  line  passing  through 
the  following  points : 

x'  =  Z,         2/ --4,     ^'  =  % 
x''=-6,     y''  =  6,        z"  =  Z. 

Ans.  icrzr— 8^4-19,    2/=10^-24 
Ex.  2.  Find  the  equations  to  the  straight  line  passing  through 
the  following  points : 

x'^^,     y'  =  -^,     z'^-Z, 
x''z=0,    y"=\,        z"=-% 
f  Ans.  x=—4:Z--8,    y=Sz^7. 

273.  To  determine  the  conditions  requisite  for  the  intersec- 
tion of  tioo  straight  lines.  Two  straight  lines  which  are  not 
parallel  must  meet  if  tliey  are  situated  in  the  same  plane,  but 


THE   STKAIGHT  LINE  IN   SPACE.  203 

this  is  not  necessarily  true  for  lines  situated  any  where  in  space. 
In  order  that  two  lines  may  meet,  there  must  be  a  particular 
relation  among  the  constant  quantities  in  their  equations.  In 
order  to  discover  this  relation,  let  the  equations  to  the  lines  be 


If  these  lines  intersect,  that  is,  have  one  point  in  common,  the 
co-ordinates  of  this  point  must  satisfy  both  sets  of  equations, 
or  for  this  point  the  values  of  x,  y,  and  3  must  be  the  same  in 
all  the  equations.  Since  x  of  the  one  line  equals  x  of  the  oth- 
er, we  have  (m  —  m^)^  -{-a—a^  =  0, 

a' —a 
or  z— -,/ 

and  since  y  of  one  line  equals  y  of  the  other,  we  have 

y-h 

or  z= ,. 

n—n 

But  z  of  the  one  line  is  equal  to  z  of  the  other;  hence 

a'^a      b'-h 


m—m     71— n 
Hence,  when  the  lines  intersect,  the  relation  between  the  con- 
stants is  given  by  the  equation 

{a'-a){n-n')  =  {y~b){m-m').  (1) 

Conversely  J  when  this  equation  exists  the  two  lines  intersect,  j?  '" 
The  cozordinates  of  the  point  of  intersection  may  be  deter- ^ 
mined  by  substituting  in  the  expressions  for  x  and  y  the  value    / 
of  ^  just  found.     They  are 

iina'—in'a            rib' —n'b 
x=: ]—,     y— T", 

a'-a      V-b 

S  = 7  = 7. 

in—m     n—n 

These  values  of  x  and  ?/,  with  either  value  of  s,  will  give  a 
point  of  intersection  when  equation  (1)  is  satisfied. 

If  m=m\  and  Qi—n',  equation  (1)  is  satisfied,  and  the  values 
of  X,  2/,  and  z  become  infinite.  The  point  of  intersection  is 
then  at  an  infinite  distance ;  that  is,  the  two  lines  are  parallel. 


204  ANALYTICAL   GEOMETRY. 

Bat  when  m  —  m!^  the  projections  of  the  two  lines  on  the 
plane  xz  are  parallel,  and  when  n=n'  the  projections  on  the 
plane  ?/s  are  parallel.  Hence,  if  two  right  lines  in  space  are 
parallel^  their  projections  on  the  same  co-ordinate  plane  will 
he  parallel. 

274.  To  find  the  equations  of  the  straight  line  which  passes 
through  a  given  point  and  is  parallel  to  a  given  line.  Let 
a?',  y',  z'  be  the  co-ordinates  of  the  given  point.  The  equations 
of  the  straight  line  passing  through  this  point  (Art.  271)  are 

x—x'  =  m{z—z'), 
and  y—y'—n{z—z'). 

In  order  that  this  line  may  be  parallel  to  a  given  line,  its 
projections  on  the  co-ordinate  planes  must  be  parallel  to  the 
projections  of  the  former  line  (Art.  273) ;  that  is,  thej  must 
cut  the  axis  of  z  at  the  same  angle.  The  quantities  m  and  n 
therefore  become  known,  and  if  we  represent  the  tangents  of 
the  given  angles  by  m'  and  ^',  we  shall  have 
x—x'—m'iz—z')., 

y-y'=n'{z-Z:\    ^ 

which  are  the  equations  of  the  required  line. 

Ex.  Find  tlie  equation  of  a  straight  line  which  passes  through 
the  point  x'  =  Z,     y'——2,     z'  —  l, 

and  is  parallel  to  the  line  whose  equations  are 
x=4:Z-{-6,     y——z-\-Z. 

21b.  To  find  the  relation  which  exists  among  the  angles 
which  any  straight  line  mahes  with  the  axes  of  co-ordinates. 
Let  a,  j3,  and  7  represent  the  angles  which  the  straight  line 
makes  with  the  axes  of  a?,  y,  and  z.  From 
the  origin,  draw  a  line  AP  parallel  to  the 
proposed  line ;  the  angles  which  it  makes 
with  the  co-ordinate  axes  will  be  the  same 
as  those  made  by  the  proposed  line.  In 
AP  take  any  point  P,  and  from  it  draw  a 
line  perpendicular  to  each  of  the  co-ordinate  planes.     In  the 


THE    STRAIGHT  LINE   IN    SPACE.  205 

triangle  APG,  right-angled  at  G,  we  have  AG=AP  cos.  7; 
also,  in  the  triangle  APF,  right-angled  at  F,  we  have  Ar= AP 
COS.  j3 ;  and  in  the  triangle  APE,  right-angled  at  E,  we  have 
AE=AP  COS.  a.     But  by  Art.  265  we  have 
AE^-fAF'+AG'=zAP; 
hence      AF  cos.  'a + AT'  cos.  =j3  -f  AP^  cos.  'y  =  AF ; 
or,  dividing  by  AP^,  we  have 

cos.^'a+cos. ''/3  +  cos.''7  =  l;  (1) 

that  is,  the  sum  of  the  squares  of  the  cosines  of  the  angles 
which  any  straight  line  makes  with  the  co-ordinate  axes  is 
equal  to  unity. 

If  it  is  required  to  determine  the  value  of  each  cosine,  let 

x—mz,    y=nz, 

be  the  equations  of  the  line  AP  (Art.  271).     Then 

COS.  a:=m  COS.  7,  and  cos.  j3=^2'  cos.  7. 

Substituting  these  values  in  equation  (1),  we  obtain 

m'  COS.  "7  -f  n^  COS.  ""y  -f  cos.  '7  =  1  ? 

whence  cos. y—    ,  - :    ^   C6-S"1L 

^     ^/m'  +  n'-{^l' 

also,  cos.a=-^=^=^===:,      ^    (c-d/. 

vm -f^ +1 


and  COS.  j3 


=.  Go 


V 


In  these  equations,  m  denotes  the  tangent  of  the  angle  which 
the  projection  of  the  proposed  line  upon  the  plane  xz  makes 
with  the  axis  of  z;  and  n  denotes  the  tangent  of  the  angle 
which  the  projection  on  the  plane  yz  makes  with  the  axis  of  z. 


206 


ANALYTICAL    GEOMETKT. 


'    SECTION  III. 


OF   THE   PLANE   IN   SPACE. 


276.  The  equation  of  a  surface  is  an  equation  which  ex- 
presses the  relation  between  the  co-ordinates  of  every  point  of 
the  surface. 

Three  points,  not  in  the  same  straight  line,  are  sufficient  to 
determine  the  position  of  a  plane  (Geom.,  Bk.  YIL,  Prop.  2, 
Cor.  1) ;  hence,  if  we  know  the  points  where  a  plane  BCD  in- 
tersects the  three  co-ordinate  axes,  the  po- 
sition of  the  plane  will  be  determined. 

The  intersections  of  any  plane  with  the 
co-ordinate  planes  are  called  its  traces. 
Thus  BC  is  the  trace  of  the  plane  BCD 
on  the  plane  XY,  BD  is  its  trace  on  the 
plane  ZX,  and  CD  is  its  trace  on  the  plane 
ZY. 


277.  To  find  the  equation  to  a  plane.  Let  AX,  AY,  AZ  be 
three  rectangular  axes,  and  let  BCD  be  the  plane  whose  equa- 
tion is  required  to  be  determined.  Let  the  plane  intersect  the 
axes  in  the  points  B,  C,  D,  and  let  AB  be  denoted  by  a^  AC  by 
J,  and  AD  by  c.     Take  any  point  P  in  the  given  plane,  and 

through  P  draw  the  plane  EGH 
parallel  to  the  co-ordinate  plane 
YZ,  and  cutting  the  given  plane 
and  the  other  co-ordinate  planes 
in  the  triangle  EGH.    Draw  PE 
perpendicular  to  the  plane  YX. 
Then  will  the  co-ordinates  of  the 
point  P  be 
jrzr  AE,  ?/=ER,  and  2=PE. 
It  is  required  to  find  an  equation  between  these  co-ordinates 
and  the  intercepts  a^  h,  and  c. 


OF  THE  PLANE  IN  SPACE.  207 

By  similar  triangles  BAG,  BEG,  we  have 
BA:  AC::  BE:  EG, 
or  a:b::a—x:^G. 

Hence  EG=:^--; 

also,  KG  =  J— ?/— — . 

Again,  by  similar  triangles  D  AC,  PEG,  we  have 
DA:AC::PR:KG, 

or  c:o\'.z\o—y— — ; 

whence  ahz=ahG—acy—hcx, 

or  lex  4-  acy + abz — abc,  (1) 

X    y    z    _^ 

or  -+|+-^1  (2)      ■ 

which  is  the  equation  of  a  plane  in  terms  of  its  intercepts  on 
the  three  axes.  This  equation  is  similar  in  form  to  the  equa- 
tion of  a  straight  line  (Art.  42).  If  we  represent  the  coefficients 
of  X,  y,  and  z  in  eq.  (1)  by  A,  B,  and  C,  this  equation  assumes 
the  form  Aaj+B^+C^+D^O,  (3) 

being  a  complete  equation  of  the  first  degree  containing  three 
variables,  and  this  is  the  form  in  which  the  equation  of  a  plane 
is  usually  written. 

278.  Having  given  the  equation  of  a  plane,  to  determine  the 
equations  of  its  traces.    Let  the  equation  of  the  plane  be 

A£c-fBy+C^+D=:0; 
then,  for  every  point  in  this  plane  which  is  situated  likewise  in 
the  plane  of  xy,  that  is,  for  every  point  in  the  trace  on  the  plane 
of  xy,  we  must  have  ^=0.    Hence  the  equation  of  this  trace  is 
Ax  +  By-\-T>  =  0.  (1) 

In  like  manner,  for  every  point  in  the  trace  on  the  plane  of 
xZj  we  must  have  2/=0 ;  hence  the  equation  of  this  trace  is 
Ax-\-Qz-\-T>  =  0.  (2) 

So  also  the  equation  of  the  trace  on  the  plane  of  yz  is 
By4-C2  +  D  =  0.  (3) 


20 S  ANALYTICAL    GEOMETRY. 

If  in  equation  (1)  we  make  2/=0,  the  resulting  value  of  x, 

viz.,  —  ^,  will  be  the  distance  from  the  origin  to  the  point  where 

the  given  plane  meets  the  axis  of  x.     If  we  make  x=0,  we 

have  y=—^  for  the  distance  from  the  origin  to  the  point 

where  the  plane  meets  the  axis  of  y.     If  in  equation  (2)  we 

make  x  =  0,  we  have  z——j^  for  the  distance  from  the  origin 

to  the  point  where  the  plane  meets  the  axis  of  z. 

If  D  =  0,  the  proposed  plane  must  pass  through  the  origin. 
Ex.  1.  Find  the  traces  of  the  plane  whose  equation  is 

2x-^y+1z—10  =  0. 
Ex.  2.  Determine  where  the  plane  whose  equation  is 
3aj+4?/+5^— 60  =  0 
meets  the  three  co-ordinate  axes. 

Ans.  x=20, 2/ =15,  s=12. 
Ex.  3.  Determine  where  the  plane  whose  equation  is 
3x-4:y+2z+l^=:0 
meets  the  three  co-ordinate  axes. 

279.  To  find  the  equation  of  the  jplane  which  passes  through 
three  given  points.  If  in  equation  (2),  Art.  277,  we  represent 
the  coefficients  of  x,  y,  and  z  by  M,  N,  and  P,  the  equation  of 
the  plane  will  become 

l£x+^y+Vz=l,  (1) 

Let  the  co-ordinates  of  the  three  given  points  be 

x',y',z';    x",y",z";     x"',y"',^"'. 
Since  the  plane  passes  through  the  three  given  points,  the 
co-ordinates  of  each  of  these  points  must  satisfy  the  equation 
of  the  plane,  so  that  we  must  have 

Ux' ^^y' +  Vz' =1, 
l^x" -^'Ny"  ■\-Vz"  =  1, 
^ix"'^-'^y'"-\^'Vz"'  =  l. 
From  these  three  equations  the  values  of  the  three  constants 
M,  K,  and  P  may  be  determined,  and  if  these  values  are  sub- 


OF   THE   PLANE   IN   SPACE.  209 

stituted  ill  equation  (1),  we  shall  have  the  equation  of  a  plane 
passing  through  the  three  given  points. 

Ex.  1.  Find  the  equation  of  the  plane  passing  through  the 
three  points 

x'  =  l,  2/'- -2,      ^--3. 

Ans.  6aj  +  ll?/-132-23  =  0. 
Ex.  2.  Find  the  equation  of  the  plane  passing  through  the 
three  points  a?'  =  3,  2/'  =  ^ ?      ^' — ^) 

x''^=-2,    y"'^l,    z'''=0, 

A7is.llx-3y-13z+25=0. 

i  280.  To  detevmine  the  conditions  which  must  subsist  in 
order  that  a  straight  line  may  bejparallel  to  aj^lane.  Let  the 
equations  of  the  straight  line  be 

x—mz-\-a^    yz=znz-\-h, 
and  let  the  equation  of  the  plane  be 

If  through  the  origin  we  draw  a  straight  line  parallel  to  the 
given  line,  its  equations  will  be 

x=mz,     y=nz; 
and  if  through  the  origin  we  also  draw  a  plane  parallel  to  the 
given  plane,  its  equation  (Art.  278)  will  be 

Aaj+B2/+C^=0. 
Kow,  if  the  first  line  be  parallel  to  the  first  plane,  the  line 
drawn  through  the  origin  must  coincide  with  the  plane  drawn 
through  the  origin ;  hence  the  co-ordinates  x  and  y  of  this 
straight  line  must  satisfy  the  equation  of  the  plane.  If  we 
substitute  the  values  of  x  and  y  in  the  equation  of  the  plane, 
we  have  Amz + B/is  -f-  C^  =  0 ; 

or,  dividing  by  z,  we  have 

Am-fB;i  +  C==0, 
which  is  the  analytical  condition  that  a  right  line  shall  be  par- 
allel to  a  plane. 


210  ANALYTICAL   GEOMKTKT, 


c 


f,  ,  281.  To  determine  the  conditions  whicJi  must  subsist  in 
drder  that  two  planes  may  he  parallel.  Let  the  equations  of 
the  two  planes  be    Aaj+By  +  C^  -f  D  =0, 

The  traces  of  these  planes  on  either  of  the  co-ordinate  planes 
must  be  parallel,  otherwise  the  two  planes  would  meet.     The 
equations  of  the  traces  on  the  plane  of  xz  (Art.  278)  are 
AD       _     A^      5^ 

2_  -  ^X-  ^,      Z-  -  ^,X-  ^,. 

If  these  traces  are  parallel,  we  must  have  . 

o~o'-  '       -^ 

Comparing  the  traces  on  the  other  co-ordinate  planes,  we  shall 
1     ^  ^  B    B^     A    A' 

also  nnd  C"!/'    B~W' 

The  last  equation  could  be  derived  from  the  two  others,  and 
hence  the  three  equations  express  but  two  independent  condi- 
tions. 

282.  If  a  straight  line  he  perpendicular  to  a  plane ^  the  pro- 
jection of  this  line  on  either  of  the  co-ordinate  planes  will  he 
perpendicular  to  the  trace  of  the  given  plane  on  that  co-ordi- 
nate plane. 

Let  MN  be  the  co-ordinate  plane,  ABCD  the  proposed  plane, 

EH  the  line  perpendicular  to  it,  and 
let  GH  be  the  projection  of  EH  on 
the  plane MN.  The  projecting  plane 
EGH  is  perpendicular  to  MN ;  and 
since  the  line  EH  is  in  the  plane 
A  ^  EGH,  the  plane  EGH  is  perpendic- 

ular to  the  plane  BD  (Geom.,  Bk.  YH.,  Prop.  6).  Hence  the 
plane  EGH  is  perpendicular  to  each  of  the  planes  MN  and  BD ; 
it  is  therefore  perpendicular  to  their  common  section  AB 
(Geom.,  Bk.VII.jProp.  8).  Hence  AB,  which  is  the  trace  of 
the  given  plane  on  the  plane  MjST,  is  perpendicular  to  the  plane 
EGH,  and  is  therefore  perpendicular  to  the  line  GH,  which  it 


OF  THE  PLANE  IN  SPACE.  211 

meets  in  that  plane  (Geom.,  Bk.YII.,  Def.  1) ;  that  is,  GH,  which 
is  the  projection  of  the  given  line  EH,  is  perpendicular  to  AB, 
which  is  the  trace  of  BD  on  the  plane  MN. 

283.  To  determine  the  conditions  which  onust  subsist  in 
order  that  a  straight  line  may  he  jperjpendicular  to  a  plane. 

Let  the  equation  of  the  plane  be 

Aa?+B?/+C^+D=:0, 
and  let  the  equations  of  the  projections  of  the  straight  line  be 

x=mz-\-a,    y—nz-\-l). 
The  equation  of  the  trace  of  the  plane  on  xz  is 
Aa;4-C^+D  =  0, 
C      D 

or  X——-irZ  —  -7-. 

A      A 

The  equation  of  the  trace  on  yz  is 

By+C^+D  =  0, 
C      D 

Bat  since  the  projections  of  the  line  must  be  perpendicular  to 
the  traces  of  the  plane  (Art.  282),  we  shall  have  (Art.  46) 

A       ,        B 

m=p-,  and  n  =  -^^ 

which  are  the  conditions  required. 

284.  To  find  the  equation  ofajplane  that  passes  through  a 
given  pointy  and  is  perpendicular  to  a  given  straight  line. 

Let  x',  y\  z'  be  the  co-ordinates  of  the  given  point,  and  let 
the  equations  to  the  given  line  be 

x=zmz-\-a,2iXidi  y—nz-^h. 
Also,  let  the  equation  of.  the  plane  be 

Aaj+By-fC2+D  =  0. 
Since  the  point  {x\  y\  z')  is  in  this  plane,  we  have 

Aaj'  +  By+C3'+r>  =  0; 
hence  A{x~x')^'B{y—y')-\-Q(z-z')=zO, 

which  is  the  equation  of  any  plane  passing  through  the  point 
{x\  y\  z').     But  by  Art.  283  we  liavc 


212  ANALYTICAL    GEOMETRY. 

A=mC,  and  B=7iC; 
hence  mC(x^x')-\-nC(y—y')-\-C{2—s')=0, 

or  m{x—x^)+n{y—y')  +  {z—z')=0, 

which  is  the  equation  required. 

285.  To  find  the  equation  of  a  straight  line  drawn  from  the 
origin  perpendicular  to  a  given  plane,  and  determine  its  length. 
Let  the  equation  of  the  given  plane  be 

Aa;+By+C5+D  =  0.  (1) 

The  equations  of  a  line  passing  through  the  origin  are 

x—mz,     y=nz. 
But  if  this  line  be  perpendicular  to  the  plane,  we  must  have 

A  B 

(Art.  283)  ^=  p>  ^^^  ''^==p  ; 

hence  the  equations  of  the  line  passing  through  the  origin  and 
perpendicular  to  the  plane  are 

^=jr,    y^-Q'  (2) 

Those  values  of  x,  y,  and  z,  Avhich,  when  taken  together,  will 
satisfy  equations  (1)  and  (2)  at  the  same  time,  must  be  the  co- 
ordinates of  a  point  common  to  the  line  and  plane ;  therefore, 
by  combining  these  equations,  and  deducing  the  values  of  x,  y, 
and  z,  we  shall  obtain  the  co-ordinates  of  the  point  in  which 
the  line  pierces  the  plane.  The  distance  of  this  point  from  the 
origin  may  then  be  found  by  Art.  265.  -t '  -I  W  '^'4:  ^ 

If  P  represent  the  length  of  the  perpendicular,  we  shall  have 

Va'^+b^+c^' 

Ex.  1.  Find  the  equations  of  a  straight  line  passing  through 
the  origin  and  perpendicular  to  the  plane  whose  equation  is 

2^-42/+^— 8  =  0. 
Find,  also,  the  point  in  which  the  line  pierces  the  plane,  and 
find  the  length  of  the  perpendicular. 

Ans.  The  equations  of  the  line  are  x—2z,      y=—4:z; 

in...  16  32  8 

it  pierces  the  plane  in  the  point  x=^,y=—:^,  ^~2T' 

8 
and  the  length  of  the  perpendicular  is  -7=. 


OF  THE  PLANE  IN  SPACE.  21S 

Ex.  2.  Find  the  length  of  the  perpendicular  from  the  origin 
upon  the  plane  whose  equation  is  ^  ^ 

286.  To  find  the  equations  of  the  iiitersection  of  two  given 
planes. 

Let  the  equations  of  the  two  planes  be 

If  the  given  planes  intersect,  the  co-ordinates  of  their  line  of 
intersection  will  satisfy  at  the  same  time  the  equations  of  both 
planes.  If,  therefore,  we  combine  the  two  equations  and  elim- 
inate z,  we  shall  obtain  an  equation  between  x  and  y,  which  is 
the  equation  of  the  projection  on  the  plane  xy  of  the  intersec- 
tion of  the  planes. 

In  a  similar  manner  we  may  find  the  equation  of  the  pro- 
jection of  the  intersection  on  the  plane  xz.  But  the  equations 
to  the  projections  of  a  line  on  two  co-ordinate  planes  are  the 
equations  to  the  line  itself ;  hence  the  two  equations  thus  found 
are  the  required  equations  to  the  intersection. 

Ex.  Find  the  equations  to  the  intersection  of  two  planes  of 
which  the  equations  are 

Zx+  y  •\-  z  +4=0. 

Ans  \  ^^^-^  ^^  +^^=^' 

,0  ■     '^ 
"'l  J 


214 


ANALYTICAL   GEOMETRY. 


SECTION  TV. 


OF   SUEFACES   OF   KEVOLrTION. 


287.  Definitions.  A  solid  of  revolution  is  a  solid  which 
may  be  generated  by  tlie  revolution  of  a  plane  surface  about  a 
fixed  axis. 

A  surface  of  revolution  is  a  surface  which  may  be  generated 
by  the  revolution  of  a  line  about  a  fixed  axis. 

The  revolving  line  is  called  the  generatrix^  and  the  line  about 
which  it  revolves  is  called  the  axis  of  the  surface  or  solid^  or 
the  axis  of  revolution.  The  section  made  by  a  plane  passing 
through  the  axis  is  called  a  meridian  section. 

It  follows  from  the  definition  that  every  section  made  by  a 
plane  perpendicular  to  the  fixed  axis  is  a  circle  whose  centre  is 
in  that  axis. 

288.  The  number  of  solids  of  revolution  is  unlimited,  but 
those  w^hich  are  of  most  frequent  use  are  the  cylinder ^  cone, 
sj>here,  spheroid, ;paraholoid,  and  hyperholoid. 

The  equation  to  a  surface  of  revolution  is  simplest  when  the 
axis  of  revolution  coincides  with  one  of  the  co-ordinate  axes. 
In  the  following  problems  we  shall  suppose  the  axis  of  revolu- 
tion to  coincide  with  the  axis  of  z,  and  the  co-ordinate  planes 
to  be  at  rio-ht  ans-les  to  each  other. 


289.  To  find  the  equation  to  the  surface 
of  a  right  cylinder.  A  right  cylinder  may 
be  supposed  to  be  generated  by  the  revolu- 
tion of  a  rectangle  about  one  of  its  sides  as 
an  axis. 

Let  CE  be  one  side  of  a  rectangle,  and  let 
it  revolve  about  the  opposite  side  AB  as  an 
axis ;  it  is  plain  that  any  point  of  CE,  as  D, 
in  its  revolution  will  describe  the  circumference  of  a  circle. 


OF   SURFACES   OF  REVOLUTION. 


215 


Let  AX,  AY,  AZ  be  the  rectangular  axes  to  which  the  cylin- 
der is  referred,  having  the  origin  at  the  centre  of  the  base  of 
the  cylinder,  and  let  the  axis  of  z  coincide  with  the  axis  of  the 
cylinder. 

Let  the  co-ordinates  of  any  point  P  on  the  surface  be  AN=^, 
NM=:a?,  and  MP==2//  ^^^^^  ^^  square  on  NP=the  sum  of  the 
squares  on  NM  and  MP,  or 

But  PIS',  which  equals  DN",  is  a  constant  quantity,  and  z  may 
have  any  value  whatever,  so  that  the  equation  of  a  right  cylin- 
der is  x'^-{-y^=c',  z  being  indeterminate. 


290.  To  find  the  equation  to  the  surface  of  a  right  cone. 
A  right  cone  may  be  supposed  to  be  generated  by  the  revolu- 
tion of  a  right-angled  triangle  about  one  of  its  perpendicular 
sides  as  an  axis,  the  hypothenuse  generating  the  curved  surface, 
and  the  remaining  perpendicular  side  generating  the  base. 

Let  AC  be  the  hypothenuse  of  a  right-angled  triangle,  and 
let  it  be  revolved  about  AB  as  an  axis ; 
then  any  point  of  AC,  as  D,  in  its  revolu- 
tion will  describe  the  circumference  of  a 
circle. 

Let  the  origin  be  placed  at  the  vertex  of 
the  cone,  and  let  the  axis  of  z  coincide  with 
the  axis  of  the  cone ;  then,  as  in  Art.  289, 
we  shall  have   'FW=:x'-\-y''. 

Let  V  represent  the  angle  BAC,  or  the 
semiangle  of  the  cone ;  then 

NP=ND=AN  tang.  CABMAN"  tang,  v; 
x'-\-y''=zz^t2ing.''v, 
which  is  the  equation  of  the  surface  of  a  right  cone. 

If  the  generatrix  AC  is  of  indefinite  length,  the  whole  sur- 
face generated  consists  of  two  symmetrical  portions,  each  of 
indefinite  extent,  lying  on  opposite  sides  of  the  vertex.  Each 
of  these  portions  is  called  a  sheet  of  the  cone. 


that^5s, 


216 


ANALYTICAL    GEOMETRY, 


291.  To  find  the  equation  to  the  surface  of  a  sjphere.  The 
sphere  is  supposed  to  be  generated  by  the  revolution  of  a  semi- 
circle about  its  diameter. 

If  the  centre  of  the  sphere  be  at  the  origin  of  co-ordinates, 
then  the  co-ordinates  of  any  point  of  the 
sphere,  as  P,  are  PM,  MN,  and  AN,  and 
Ave  have 

DK'=.PN^=NM'+MF; 
also, 

AD=  =  AN'+ND"=m^+MF-fAN^ 
Hence,  putting  r  for  AD,  the  radius  of 
the  sphere,  we  have 

which  is  tlie  equation  of  the  surface  of  a  sphere. 


292.  To  find  the  equation  to  the  surface  of  a  jprolate  sphe- 
roid. Spheroids  are  either  prolate  or  oblate.  A  prolate  sphe- 
roid is  supposed  to  be  generated  by  the  revolution  of  an  ellipse 
about  its  transverse  axis.  An  oblate  spheroid  is  supposed  to 
be  generated  by  the  revolution  of  an  ellipse  about  its  conjugate 
axis. 

Let  BCE  be  an  ellipse,  and  let  it  be  revolved  about  its  trans- 
verse axis ;  then  any  point  of  the  circum- 
ference, as  D,  in  its  revolution  will  de- 
scribe the  circumference  of  a  circle. 

Let  the  origin  be  placed  at  the  centre  of 
the  sjpheroid.  The  equation  of  an  ellipse 
(Art.  121)  is    aY-\-lfx''  =  a'h\ 

,     a'b'-h'x' 
or  y  =— ^r— , 

where  x  represents  AIS",  which  is  now  to 
be  represented  by  z,  and  y  represents  ND, 
the  radius  of  the  circle  described  by  the 
point  D  in  its  revolution. 

Hence  ND-^^^^. 


OF    SURFACES   OF   REVOLUTION. 


217 


But                 lSrD»=NF=NM^+MP=aj'+2/'; 
hence  aj  +?/  — ^^ , 

or  a\x'^^f)-\-¥z'=a''b\ 

whicli  is  tlie  equation  of  the  surface  of  a  prolate  spheroid, 

where  a  is  supposed  to  be  greater  than  h. 


293.  To  find  the  equation  to  the  surface  of  an  oblate  sjphe- 
roid. 

Let  the  ellipse  CBE  be  revolved  about  its  conjugate  axis 
CE ;  the  point  D  in  its  revohition 
will  describe  the  circumference  of  a 
circle.   The  equation  of  an  ellipse  is 


a'l/ 


■  aY 


z 

c 

--^^ 

^-—^ 

/^^^      N 

^r 

XD 

'V.^^^ 

\ 

4- 

-^^ 

\B      ^ 

v7 

A 

) 

where  y  represents  AN,  which  is 
now  to  be  represented  by  z,  and  x 
represents  ND,  the  radius  of  the      T 
circle  described  by  the  point  D  in  its  revolution. 

Hence  nD^^^^_=^. 

But  ND'=NP=N]Vr+MF=ic^+2/% 

,.    ,     a'¥-a'3' 
hence  x  -{-y  = 75 , 

or  bl{x'  +  y')  +  a'3'  =  a'h% 

which  is  the  equation  of  the  surface  of  an  oblate  spheroid. 
The  equation  of  the  prolate  spheroid  is  sometimes  written 


x'    y'    z' 
2-t-7.2-r-^2- 


and  that  of  the  oblate  spheroid, 


1, 


a^'^a^'^b'-^' 


In  both  cases  a  is  supposed  greater  than  b. 

If  in  the  equation  of  either  spheroid  we  make  ^— <^,  we  shall 
have  aj'+y'+s'i^r', 

Avhich  is  the  equation  of  the  surface  of  a  sphere  (Art.  291). 

K 


218 


ANALYTICAL    GEOMETRY. 


294.  To  find  the  equation  to  the  surface  of  a  jparaboloid. 
A  paraboloid  is  supposed  to  be  generated  by  the  revolution  of 
a  parabola  about  its  axis. 

Let  EAC  be  a  parabola,  and  let  it  be  revolved  about  its  axis 
AB ;  then  any  point  on  the  curve,  as  D, 
in  its  revolution  will  describe  the  circum- 
ference of  a  circle.  Let  the  origin  be 
placed  at  the  vertex  of  the  parabola,  and 
let  the  axis  of  the  parabola  be  the  axis  of  z. 
The  equation  of  a  parabola  (Art.  85)  is 
if—4:ax, 

t/  where  x  represents  AN",  which  is  now  to 

be  represented  by  z,  and  y  represents  ND. 

Hence  ^D''=4:az. 

But  ND'=NF=NM^+MF=aj'+2/''; 

hence  we  have  x^-\-y'^—4:az, 

which  is  the  equation  of  the  surface  of  a  paraboloid. 


295.  To  find  the  equation  to  the  sic? face  of  an  hyperholoid. 
An  hyperboloid  is  supposed  to  be  generated  by  the  revolution 
of  an  hyperbola  about  one  of  its  axes. 

1st.  We  will  suppose  the  hyperbola  to  revolve  about  its  trans- 
verse axis.  Let  CBD  be  an  hyperbo- 
la, and  let  it  be  revolved  about  its 
transverse  axis  3E ;  then  any  point 
on  the  curve,  as  D,  in  its  revolution 
will  describe  the  circumference  of  a 
circle.  Let  the  origin  be  placed  at 
the  centre  of  the  hyperbola,  and  let 
the  transveree  axis  of  the  hyperbola 
be  the  axis  of  z. 

The  equation  of  an  hyperbola  (Art.  170)  is 

b^ 

where  x  represents  AN",  which  is  now  to  be  represented  by  z, 
and  y  represents  ND. 


OF   fiUKFACES   OF   EEVOLUTION. 


219 


Hence 

But 

hence 


or 


which  is  the  equation  of  the  surface  generated  by  revolving  an 
hyperbola  about  its  transverse  axis.  If  we  suppose  both  branch- 
es of  the  hyperbola  to  revolve,  there  will  be  generated  two  sur- 
faces entirely  symmetrical  with  respect  to  each  other.  This  is 
therefore  called  the  hyperboloid  of  revolution  of  two  sheets, 
since  it  forms  two  surfaces  entirely  separate  from  each  other. 

If  the  asymptotes  of  the  hyperbola  also  revolve  around  the 
transverse  axis,  they  will  describe  the  surface  of  a  cone  with 
two  sheets.  The  surface  of  this  cone  will  approach  the  surface 
of  the  hyperboloid,  and  will  become  tangent  to  it  at  an  infinite 
distance  from  the  centre. 

2d.  We  will  suppose  the  hyperbola  to  revolve  ahottt  its  con- 
jugate axis.  Let  CBD  be  an 
hyperbola,  and  let  it  be  revolved 
about  its  conjugate  axis  AE; 
then  any  point  on  the  curve,  as 
D,  in  its  revolution  will  describe 
the  circumference  of  a  circle. 
Let  the  origin  be  placed  at  the 
centre  of  the  hyperbola,  and  let 
the  conjugate  axis  of  the  hyperbola  be  the  axis  of  z. 

The  equation  of  the  hyperbola  is 

X-  J,  , 

where  y  represents  AK,  which  is  now  to  be  represented  by  2^ 

and  X  represents  ND. 

XT      ^                          TVT-r.2    a^^'+a^h'' 
Hence  ND'= v^ . 

But  ND==]S^F=:N^M'+MP=ic'4-2/'; 

hence  x'-\-y''= Ta , 


220  ANALYTICAL   GEOMETRY. 

or  a  V  -  h'ix' + 2/')  ==  -  a'h\ 

which  is  the  equation  of  the  surface  generated  by  revolving  an 
hyperbola  about  its  conjugate  axis.  As  both  branches  of  the 
hyperbola  are  symmetrical  with  respect  to  the  conjugate  axis, 
each  branch  in  its  revolution  will  describe  the  same  surface. 
This  is  therefore  called  the  hyperboloid  of  revolution  of  one 
sheet,  since  it  forms  one  uninterrupted  surface. 

The  equations  of  the  two  hyperboloids  of  revolution  are 

sometimes  written  —^—ti—u—'^, 

a     0      0^ 

^"     «'     f     . 
and  __+_+_^l, 

where  the  minus  sign  in  each  case  corresponds  to  an  axis  that 
does  not  meet  the  surface. 

296.  To  determine  the  curve  which  results  from  the  inter^ 
section  of  a  sjphere  loith  a  plane.  Let  d  represent  the  distance 
of  the  intersecting  plane  from  the  centre  of  the  sphere ;  let  the 
origin  be  at  the  centre  of  the  sphere,  and  let  one  of  the  co-or- 
dinate planes,  as  the  plane  of  xy,  be  parallel  to  the  cutting 
plane ;  then  every  point  in  the  intersecting  plane  will  be  given 
by  the  equation  z=d,  and  we  must  have 

x'-^7/'-\-d/=^r\ 
or  x'-^y'^^r'-d', 

which  represents  all  the  points  on  the  surface  of  the  sphere 
which  are  also  common  to  the  plane.  This  equation  represents 
a  circle  whose  radius  is  Vr''—d^,  Hence  every  section  of  a 
sphere  made  by  a  plane  is  a  circle. 

Ex.  A  sphere  whose  radius  is  10  inches  is  cut  by  a  plane 
whose  distance  from  the  centre  of  the  sphere  is  6  inches.  De- 
termine the  radius  of  the  section.  .       ,  i.  r-^ 

297.  To  determine  the  curve  which  results  from  the  inter- 
section of  a  right  cylinder  with  a  jplane.  Every  section  of  a 
right  cylinder  made  by  a  plane  parallel  to  the  base  is  a  circle ; 
we  will  therefore  suppose  the  section  to  be  made  by  a  plane 


OF    SUEFACES    OF    REVOLUTION. 


221 


inclined  to  tlie  base.  Let  APB  be  sncli  a  section,  and  let  ABC 
be  a  section  of  the  cylinder  through  its  axis,  and  perpendicular 
to  the  plane  of  the  former  section.  Draw  a 
plane  perpendicular  to  the  axis  of  the  cylin- 
der, intersecting  the  cylinder  in  a  circle  whose 
diameter  is  DE,  and  intersecting  the  first 
plane  in  PM,  which  will  therefore  be  perpen- 
dicular both  to  AB  and  DE,  and  will  be  an 
ordinate  common  to  the  section  and  the  circle. 

Let  AM=x,  FM.=y,  AB  =  2^,  AC  =  2r;  then  BM=2a-x. 

We  have  ^/^.^DM .  ME  (Geom.,  Bk.  lY.,  Prob.  23,  Cor.);  but 
by  similar  triangles  we  have 


rx 


also 
Whence 


AB :  AC : :  AM :  MD,  whence  MD =— ; 


AB :  AC : :  BM :  ME,  whence  ME=-(2^. 


■X). 


y"=-i{^ax-x% 


a 


which  is  the  equation  of  an  ellipse  (Art.  129). 

Hence  every  section  of  a  right  cylinder  made  by  a  plane  in- 
clined to  its  base  is  an  ellipse. 

Ex.  A  right  cylinder  whose  diameter  is  10  inches,  is  cut  by 
a  plane  making  an  angle  of  30°  with  the  axis  of  the  cylinder. 
Determine  the  equation  of  the  elliptic  section. 

298.  To  determine  the  curve 
which  results  from  the  interseg- 
Hon  of  a  right  cone  with  ajplane. 
Let  VBGC  be  a  right  cone,Y  the 
vertex, YH  the  axis,  andBGC  the 
circular  base.  Let  AP  be  the  line 
in  which  the  cutting  plane  meets 
the  surface  of  the  cone,  and  let 
YBHC  be  a  plane  passing  through 
the  axis  YH,  and  perpendicular 
to  the  cutting  plane  AMP.  AM, 
the  intei-section  of  these  planes, 


222  ANALYTICAL    GEOMETEY. 

is  a  straight  line ;  and,  since  the  curve  is  symmetrical  with  re- 
gard to  it,  it  is  called  the  axis  of  the  conic  section. 

Let  DPE  be  a  section  parallel  to  the  base ;  it  will  be  a  circle, 
and  DME,  its  intersection  with  the  plane  YBHC,  will  be  a 
diameter. 

Since  the  plane  DPE  and  the  plane  PAM  are  both  perpen- 
dicular to  the  plane  YBHC,  MP,  the  intersection  of  the  two 
former,  is  perpendicular  to  the  third  plane,  and  therefore  to 
every  straight  line  in  that  plane.  It  is  therefore  perpendicu- 
lar to  DE  and  to  AM.  Draw  AF  parallel  to  DE,  and  ML 
parallel  to  YB,  and  let  it  meet  YC  in  N. 

Let  AM=(ZJ,  PM=:?/,  YA=a;  let  the  angle  CYHr:=j3,  and 

the  angle  YAM,  which  is  the  inclination  of  the  cutting  plane  to 

the  side  of  the  cone,=:0;  tlien  the  angle  AMK= 180° -0-2/3. 

£C  sm  0 

Now  AM :  ME : :  sin.  AEM :  sin.  MAE,  whence  ME = ^ ; 

'  COS.  p  ' 

also,  DM=rL=AF-AL=:2^  sin.  j3-AL,  and 

AM :  AL : :  sin.  ALM :  sin.  AML,  whence  AL^'^^'^'^^t^^^ ; 

'  cos.j3       ' 

therefore  T>M.=2a  sin.  B-x  ^'"'  ^^"tf^l 

'^  cos.  /3 

But  by  Geom.,  Bk.  lY.,  Prob.  23, 

MF=DM.ME; 

/  X  sin.  0  i         '     o     ^  ^^"-  (^  +  ^0)  I         n\ 

hence  y  = jr  \  2^  sm.H  — r,         } ,      (1) 

^       cos.j3   I  ^  cos.j3         )  '     ^  ^ 

which  is  the  equation  of  the  curve  resulting  from  the  intersec- 
tion of  the  cone  by  a  plane. 

Comparing  this  equation  with  the  equation  if^rnx-^-nx^ 
(Art.  234),  which  represents  an  ellipse,  hyperbola,  or  parabola, 
according  as  n  is  negative,  positive,  or  zero,  we  see  that  the  sec- 
tion is  an  ellipse,  hyperbola,  or  parabola  according  as  the  co- 
efficient of  the  last  terra  of  the  equation  is  negative,  positive, 
or  zero.  In  order  to  investigate  these  cases,  w^e  will  suppose 
the  cutting  plane  to  turn  about  A,  so  as  to  make  all  possible 
angles  with  the  side  of  the  cone. 


OF  SURFACES  OF  REVOLUTION.  223 

299.  Discussion  of  the  equation  to  a  conic  section.  Equa- 
tion (1)  of  Art.  298  will  represent  in  succession  every  line 
which  it  is  possible  to  cut  from  a  given  right  cone  by  a  plane, 
if  we  suppose  /3  to  remain  unchanged,  while  all  values  are  as- 
signed to  Q  from  0  to  180°,  and  all  values  to  a  from  0  to  in- 
finity. 

Case  first  Let  Q=0\  then  equation  (1)  reduces  to  y'  =  0. 
This  is  the  equation  to  the  straight  line  which  is  the  axis  of  x, 
and  we  see  from  the  figure  that  when  Q  —  0  the  cutting  plane 
becomes  tangent  to  the  cone,  and  the  line  AM  coincides  with 
AY.  In  this  case  the  section  is  said  to  be  a  straight  line. 
The  same  case  occurs  when  0  =  180°. 

Case  second.  Let  0+2/3  <180° ;  then  sin.  (0-j-  2/3)  will  be  pos- 
itive ;  moreover,  sin.  0  is  positive  so  long  as  Q  is  supposed  to  be 
comprised  between  0  and  180°,  and  cos.^/3  is  necessarily  posi- 

sin.  0  sin.  (0  +  2/3)  .  .  ,  .       ,,, 

tive;  nence  — ^ is  negative,  and  equation  (1) 

assumes  the  form  y^  —  mx—nx'^ 

which  is  the  equation  of  an  ellipse.  We  see  from  the  figure 
that  in  this  case  the  angles  YAM  and  AYF,  or  ANM,  are  to- 
gether less  than  180°;  hence  the  lines  YF  and  AM,  if  pro- 
duced indefinitely  towards  the  base  of  the  cone,  will  meet ; 
that  is,  the  sectional  plane  cuts  both  sides  of  the  cone.  Hence 
the  section  is  cm  ellipse  when  the  cutting  plane  meets  hoth 
sides  of  the  cone.     See  fig.  Art.  301. 

Case  third.  In  the  preceding  case  the  angle  0  may  be  equal 
to  the  angle  YAF,  or  90°-/3,in  which  case  0  +  2/3  =  9O°+/3, 
and  equation  (1)  reduces  to  y'^  =  2ax  sin.  j3— a;'*,  which  is  the 
equation  of  a  circle  (Art.  63).  We  see  that  in  this  case  the 
cutting  plane  is  parallel  to  the  base,  and  hence  the  ellipse  he- 
comes  a  circle  when  the  cutting  plane  is  parallel  to  the  base 
of  the  cone. 

Case  fourth.  Let  0+2/3  =  180°;  in  this  case,  sin.  (0+2/3)  =  0, 
and  equation  (1)  becomes 

y"  =  2ax  sin.  0  tang.  /3, 
which  is  the  equation  of  a  parabola  (Art.  85).     We  see  that  in 


224  ANALYTICAL   GEOMETRY. 

this  case  180°— 0— 2j3  =  0 ;  that  is,  the  angle  AMN"  is  zero,  or 
the  cutting  plane  is  parallel  to  the  side  of  the  cone.  Hence  the 
section  heco7nes  a  parabola  when  the  cutting  plane  and  the  side 
of  the  cone  make  equal  angles  with  the  base  (see  fig.,  Art.  SOI). 

Case  fifth.  Let  0  +  2j3>18O°;  then  sin.  (0+2/3)  will  be  neg- 
ative, and  —  sin.  (0+2j3)  will  be  positive,  and  equation  (1)  as- 
sumes the  form  y^—inx+nx", 

which  is  the  equation  of  an  hyperbola.  We  see  from  the  fig- 
ure that  in  this  case  the  angles  YAM  and  AKM  are  together 
greater  than  180° ;  hence  the  lines  YB  and  AM,  though  pro- 
duced indefinitely  towards  the  base  of  the  cone,  will  not  meet, 
but  if  these  lines  are  produced  in  the  opposite  direction  they 
will  meet ;  that  is,  the  cutting  plane  intersects  both  cones,  and 
the  curve  consists  of  two  branches,  one  on  the  surface  of  each 
cone. 

When  0=180°,  the  line  AM  produced  returns  to  the  same 
position  which  it  had  when  0=0 ;  and  when  0  becomes  greater 
than  180°,  the  line  AM  assumes  the  same  positions  already  de- 
scribed. W"e  therefore  obtain  all  the  possiblo  positions  of  the 
line  AM  by  supposing  0  to  be  comprised  between  the  limits 
0  and  180°. 

SOO,  liesidt  of  a  change  in  the  value  of  a.  The  preceding 
results  remain  unchanged  so  long  as  a  remains  finite.  When 
a  becomes  zero,  the  cutting  plane  passes  through  Y,  the  vertex 
of  the  cone,  and  equation  (1)  becomes 

sin.eBin.(e+23)^. 
^  cos.  p  ^  ^ 

This  equation  furnishes  three  cases : 

Case  first.  Let  0+2j3<18O°;  then  -sin.  (0  +  2)3)  will  be 
negative.  In  this  case  equation  (2)  can  only  be  satisfied  when 
ic=0,  ?/=0,  which  are  the  equations  of  appoint.  A  point  is 
then  to  be  regarded  as  a  particular  case  of  the  ellipse.  This 
case  happens  when  the  cutting  plane,  passing  through  the  ver- 
tex Y,  occupies  a  position  within  the  angle  BYC^ 

Case  second.  Let  0  +  2/3  =  180°;  then  sin.  (0  +  2/3)  =  O,  and 


OF   SUEFACES   OF   KEVOLUTION. 


225 


equation  (2)  reduces  to  y'^^O.  The  section  then  becomes  a 
straight  line,  or  it  may  be  regarded  as  a  double  line,  since  the 
equation  may  be  written  ?/=  ±0. .  A  straight  line  (or  a  double 
line)  is  then  a  particular  case  of  the  parabola. 

Case  third.  Let  0  +  2j3>18O°;  then  -sin.  (0+2j3)  will  be 
positive,  and  equation  (2)  assumes  the  form 

2/=  -^;^^-'^''-  ^  si"- (^+2/3), 

which  represents  two  intersecting  straight  lines.  This  case 
happens  when  the  straight  line  AM,  passing  through  the  ver- 
tex y,  meets  BC  between  the  points  B  and  C.  The  cutting 
plane  then  meets  the  surface  of  the  cone  in  two  straight  lines 
which  pass  through  Y.  Two  intersecting  straight  lines  are 
then  to  be  regarded  as  a  particular  case  of  the  hyperbola. 


301.  Hesults  of  the  preceding  discussion.  It  appears 
the  preceding  investigation  that  if  a  right  cone  be  cut 
plane,  the  section  will  be 

(1)  K  parabola  when  the  plane  makes 
an  angle  with  the  axis  equal  to  half  the 
vertical  angle  of  the  cone. 

The  particular  case  is  a  double  line. 

(2)  An  ellipse  when  the  plane  cuts 
only  one  sheet  of  the  cone. 

The  particular  cases  are  a  point  and  a 
circle. 

(3)  An  hyperbola  when  the  plane  cuts 
both  sheets  of  the  cone. 

The  particular  case  is  two  straight 
lines  which  intersect  one  another. 


from 
by  a 


302.  To  determine  the  curve  which  results  from  the  inter- 
section of  any  surface  of  revolution  by  a  plane.  The  sections 
of  a  surface  made  by  the  co-ordinate  planes  are  called  the  prin- 
cipal sections  of  the  surface,  and  the  boundaries  of  the  princi- 
pal sections  are  called  the  traces  of  the  surface  on  the  co-ordi- 

E2 


226 


ANALYTICAL   GEOMETRr. 


nate  planes.  The  equation  to  a  trace  is  determined  by  putting 
the  ordinate  perpendicular  to  the  plane  of  the  trace  =0  in  the 
general  equation  (Art.  278).  If,  then,  the  cutting  plane  coin- 
cided with  one  of  the  co-ordinate  planes,  we  could  easily  find* 
the  trace  of  the  given  surface  upon  that  plane,  and  this  would 
be  the  required  curve  of  intersection.  We  may  make  the  cut- 
ting plane  coincide  with  one  of  the  co-ordinate  planes  by  a 
transformation  of  the  co-ordinates.  In  the  case  of  a  surface 
of  revolution,  we  may  proceed  as  follows : 

Through  AZ,  the  axis  of  revolution,  draw  a  plane  perpen- 
dicular to  the  proposed  section,  and 
call  this  the  plane  xz^  the  origin  be- 
ing at  A  in  the  plane  XAZ.  Let 
AX^  represent  the  intersection  of  the 
cutting  plane  with  the  plane  xz.  The 
lines  AX'  and  AY  will  then  be  per- 
pendicular to  each  other,  and  both 
will  be  in  the  cutting  plane. 

Let  P  be  any  point  of  the  curve  of  intersection,  and  from  P 
draw  PM  perpendicular  to  the  plane  xy^  and  from  M  draw  MN 
perpendicular  to  AX.  The  co-ordinates  of  P  referred  to  tlie 
primitive  axes  are 

aj=AK,  .2/=MN,  ^==PM. 
Let  the  point  P  be  now  referred  to  the  two  axes  AX',  AY, 
which  are  in  the  plane  of  the  given  section.  Through  P  draw 
PR  perpendicular  to  AY,  and  join  MR  The  angle  PRM, 
which  we  w411  denote  by  0,  is  the  angle  which  the  cutting 
plane  makes  with  the  plane  xy.  The  co-ordinates  of  P  referred 
to  the  new  axes  are 

aj'^PE,    2/'=:AR=MN,    ^'=0. 
In  the  right-angled  triangle  PMR  we  have 

EM^AN^PRcos.PRM,    or  aj=aj' cos.0, 
PM=  PR  sin.  PRM,     or  z^x'  sin.  0 ; 

also  we  have  MNi=AR,     ory—y'. 

If  the  origin  be  changed  to  a  point  in  the  plane  xz  whose  co- 


ordinates are 


x=a, 


y 


0, 


z=c, 


OF  SURFACES  OF  REVOLUTION.  227 

these  equations  become    aj=«^+aj'  cos.  0, 

z=c^+  x'  sin.  %. 
If  these  values  be  substituted  for  ic,  ?/,  and  z  in  the  equation 
of  the  given  surface,  the  result  can  only  belong  to  points  com- 
mon to  the  surface  and  the  cutting  plane,  and  will  therefore 
represent  the  required  curve  of  intersection. 

303.  To  determine  the  curve  of  intersection  of  a  plane  and 
a  prolate  spheroid. 

The  equation  of  the  given  surface  (Art.  292)  is 

(Jb 

Substituting  the  values  of  x^  ?/,  and  z  found  in  Art.  302,  this 
equation  becomes 

{a,  4-  ^  COS.  %y + 2/' + -ic,  +  X  sin.  Q^  =  h\ 
a 

72  72 

or         ajXcos.'0+— sin.'0)  +  2/H2aj(-Tsin.0+^,cos.0) 
a  a 

=i^-a;-%.  (1) 

'  a 
Suppose  now  the  origin  to  be  placed  on  the  surface  of  the 
spheroid,  and  in  the  plane  xz.  The  section  of.  the  spheroid  by 
the  plane  xz  is  equal  to  the  generating  ellipse ;  hence  the  co- 
ordinates of  the  origin  must  satisfy  the  equation  of  the  ellipse ; 
that  is,  we  must  have 

.      a'a;-\-h''c;=a'b\ 

or  J»_^;_^=.0. 

'       a 

The  second  member  of  equation  (1)  reduces  therefore  to 
zero,  and  the  equation  is  of  the  form 

y^—mx—n^^ 
and  therefore  represents  an  elli2:)se.     If  0=0,  the  equation  be- 
comes y^=i'^ax—x^^ 
which  is  the  equation  of  a  circle.  .     : 

Hence  every  section  of  a  prolate  spheroid  by  a  plane  is  an 


228  ANALYTICAL   GEOMETRY. 

ellipse,  except  when  the  cutting  plane  is  perpendicular  to  the 
axis  of  revolution,  when  the  section  becomes  a  circle. 

The  same  is  true  of  the  sections  of  an  oblate  spheroid. 

Ex.  The  two  axes  of  a  prolate  spheroid  are  8  and  6,  and  the 
spheroid  is  cut  by  a  plane  passing  through  the  extremities  of 
the  axes,  and  perpendicular  to  their  plane.  Required  the  axes 
of  the  curve  of  intersection.  A^is.  5  and  3^/2. 

304.  To  determine  the  curve  of  intersection  of  a  plane  and 
a  paraboloid  of  revolution.  The  equation  of  the  given  surface 
(Art.  294)  is    '  aj' + y' = ^az. 

Substituting  the  values  of  x^  y,  and  z  given  in  Art.  302,  this 
equation  becomes 

{a^  -\-  X  COS.  &f  -{-y^= ^a{Cj  +  x  sin.  Q), 
or         x"  cos.'0-[-2/^4-(2<3^/  cos.  0—4^  sin.  Q)x=4iaG^—a^,         (1) 

Suppose  now  the  origin  to  be  placed  on  the  surface  of  the 
paraboloid,  and  in  the  plane  xz;  the  co-ordinates  of  the  origin 
must  satisfy  the  equation  of  the  generating  parabola,  and  we 
must  have  a^^^iac^^     ov  4:ac^—a^=^0. 

Equation  (1)  therefore  reduces  to  the  form 

if=nix—nx', 
and  generally  represents  an  ellipse.     If  0=0,  the  equation  be- 
comes .  y^  ~  2ax — x"^, 
which  is  the  equation  of  a  circle. 

If  0=90°,  the  equation  becomes 
y''=4:ax, 
which  is  the  equation  of  a  parabola.  Hence  the  section  of  the 
paraboloid  by  a  plane  is  a  parabola,  when  the  plane  is  parallel 
to  the  axis  of  revolution ;  it  is  a  circle  when  the  plane  is  per- 
pendicular to  this  axis ;  and  in  all  other  positions  of  the  cutting 
plane  the  section  is  an  ellipse. 

Ex.  A  paraboloid  whose  axis  of  revolution  is  45-?-,  and  its  base, 
or  greatest  double  ordinate,  32,  is  cut  by  a  plane  passing  through 
the  edge  of  the  base,  and  meeting  the  opposite  side  of  the  solid 
at  the  height  of  20  above  the  base.  Required  the  axes  of  tlie 
section.  Ans.  34.4  and  28. 


OF  SURFACES  OF  EEVOLUTION.  229 

305.  To  determhie  the  curve  of  intersection  of  a  plane  and 
an  hyperboloid  of  revolution.  We  will  suppose  the  solid  to  bo 
the  hyjperholoid  of  two  sheets  (Art.  295).     The  equation  of  the 

given  surface  is  x'-^y'— —z"  =  —  5\ 

Substituting  the  values  of  x^  y,  and  s  given  in  Art.  302,  this 
equation  becomes 

{a^-\-x  COS.  &f^-f—lc^+x  sin.  ey  =  —h\ 
a 

V  ¥c 

or        icXcos.'^— ,  sin.=0)+2/'~2ir(-/sin.  d-a,  cos.0) 

^^_5^_«;.  ■    (1) 

a  ^  ^  ^ 

If  we  place  the  origin  on  the  surface  of  the  hyperboloid,  and 
in  the  plane  xz^  the  second  member  of  this  equation  reduces  to 
zero,  and  the  equation  is  of  the  form 

y'^=z'inx—nx^. 
If  0  =  0,  the  equation  becomes 

y^  —  ^ax—x^^ 
which  is  the  equation  of  a  circle. 
If  0=90°,  the  equation  becomes 

f~{x^-\-^c,x\ 

which  is  the  equation  of  an  hyperbola. 

If  tang.  0=T,  the  equation  reduces  to 

?/'  =  9jx{Cj  cos.  0  cotang.  0 — ^^  cos.  0), 
which  is  the  equation  of  a  parabola. 

If  tang. '0<Tj,  the  curve  is  an  ellipse;  if  tang.  ^0>^,  the 

curve  is  an  hyperbola. 

In  every  case  the  section  of  the  hyperboloid  by  a  plane  is 
similar  to  the  corresponding  section  of  the  cone  formed  by  the 
revolution  of  the  asymptotes  of  the  hyperbola  (Art.  295). 


230  ANALYTICAL   GEOMETKY. 

306.  Summary  of  the  preceding  results.    The  equation  to 
the  surface  of  an  oblate  spheroid  (Art.  293)  may  be  written 

^'    y"    ^"    . 

^+^+P=l'  (1) 

and  that  of  a  prolate  spheroid, 

r^  qfl  2/^ 

The  equation  to  the  surface  of  an  hjperboloid  of  one  sheet 

r'     9/^     ^ 
(Art.  295)  is  ^+^-j5=l,  (3) 

and  that  of  an  hyperboloid  of  two  sheets  is 

^+^-^—1  (4) 

The  equation  to  the  surface  of  a  right  cone  (Art.  290)  is 
a?^ + ^z'* — ^"^  tang. '''y  =  0 ; 

if  we  divide  Dy  a",  and  put  H"  for  - — ^^i-,  the  equation  becomes 

2  5  9 

SR      ?y      z 

-,+|-p=0.  (5) 

The  equation  to  the  surface  of  a  paraboloid  (Art.  294)  is 

if  we  divide  by  a",  and  put  h  for  7,  the  equation  becomes 

-^4-^-7=0.  (6) 

a     a     0  ^  ^ 

In  each  of  these  six  equations  the  coefficients  of  x^  and  'if 

are  equal,  which  shows  that  for  each  of  these  solids  a  section 

perpendicular  to  the  axis  of  ^  is  a  circle. 

307.  More  general  form  of  the  preceding  equations.     If  we 

suppose  the  coefficients  of  x^  and  Tf  in  either  of  these  equations 

to  be  unequal,  we  shall  liave  a  new  equation  similar  in  form  to 

the  preceding,  but  representing  a  more  complex  surface.     The 

x^    u"    z^ 
equation  -j^^-j^-^\  (1) 

j'cpresents  a  surface  similar  in  some  respects  to  tliat  of  the 


OF  SDEFACES  OF  REVOLUTION.  231 

spheroid,  but  its  intersection  with  a  plane  perpendicular  to  the 
axis  of  z  is  an  ellipse  instead  of  a  circle.  All  sections  made 
by  parallel  planes  are  similar  ellipses,  and  the  surface  is  closed 
in  every  direction.  This  solid  is  called  an  ellipsoid,  and  has 
three  unequal  axes.  When  two  of  the  axes  are  equal  to  each 
other  it  is  called  an  ellipsoid  of  revolution,  because  it  may  be 
generated  by  the  revolution  of  an  ellipse  about  one  of  its  axes. 

rjffl      y-^      ^ 

The  equation  '^+^—-,  =  1  (2) 

represents  a  surface  like  the  hyperboloid  of  one  sheet,  except 
that  the  sections  perpendicular  to  the  axis  of  z  are  ellipses  in- 
stead  of  circles. 

^2       ^a       ^2 

So  also  the  equation   — +73— -i=— 1  (3) 

represents  a  surface  like  the  hyperboloid  of  two  sheets,  but  the 
sections  perpendicular  to  the  axis  of  z  are  ellipses. 

The  equation  "i+t^— ";^  =  0  (4) 

represents  a  conical  surface,  but  the  cone  has  an  elliptic  base 
instead  of  a  circular  one. 

The  equation  — +tj— -  =  0  (5) 

represents  a  surface  like  the  paraboloid  of  revolution,  except 
that  a  section  perpendicular  to  the  axis  of  z  is  an  ellipse  in- 
stead of  a  circle.     This  solid  is  called  an  elliptic  paraboloid. 

Each  of  these  surfaces  may  be  conceived  to  be  derived  from 
the  corresponding  surface  of  revolution  by  increasing  or  dimin- 
ishing the  values  of  y  in  a  constant  ratio,  in  the  same  manner  as 
oblate  and  prolate  spheroids  may  be  derived  from  the  sphere 
by  multiplying  the  values  of  3/  by  a  constant  factor,  or  as  the 
ellipse  may  be  derived  from  the  circle  by  multiplying  the  values 
of  y  by  a  constant  factor. 

308.  Surface  of  a  cone  asymptotic.    The  conical  surface 

represented  by  the  equation 

^2    ^,9     ^ 
X     y     z      ^ 


232  ANALYTICAL  'GEOMETRY. 

is  asymptotic  on  the  one  side  to  the  hyperboloid  of  one  sheet 

iC'  11^  ^ 

whose  equation  is  -j+tj— -3  =  1, 

and  on  the  other  side  to  the  hyperboloid  of  two  sheets  whose 

rji?       y^        ^ 

equation  is  -j+t?— ^  =  — 1. 

O/       0       c 

There  is  also  a  similar  relation  between  the  equations  of  two 
conjugate  hyperbolas  and  the  equation  of  their  asymptotes. 
The  equation  of  an  hyperbola  (Art.  170)  may  be  written 

and  the  equation  of  its  conjugate  hyperbola  (Art.  179)  is 
^    ^_     1 

while  the  equation  of  their  asymptotes  (Art.  214)  is 

or  -^-|5=0. 

a     0 


// 


GENERAL   EQUATION   OF   THE   SECOND   DEGREE,  ETC.         233 


SECTION  Y. 

GENEEAIi   EQUATION   OF   THE    SECOND   DEGREE   BETWEEN   THREE 

VARIABLES. 

309.  The  general  equation  of  the  second  degree  between 
three  variables  is  of  the  form 

ax^ + l^y + ^/  +  dz^  +  ^^^  -^-fy^  -\-gx-[-hy-\-7cz-\-l—0.  (1) 
We  may  transform  this  equation  into  another,  in  which  the 
axis  of  3  remains  unchanged,  by  employing  the  equations  of 
transformation  for  plane  co-ordinates  (Art.  65),  and  we  shall 
have  z=z' 

x=x'  cos.  %^y'  sin.  0 
y—x'  sin.  Q-^y'  cos.  0. 
If  we  substitute  these  values  of  the  variables  in  equation  (1), 
the  only  terms  in  the  resulting  equation  which  can  contain  the 
product  x'y'  will  come  from  the  three  terms  ax^-\-hxy+cy''. 
The  term  containing  xy  may  therefore  always  be  made  to  dis- 
appear from  equation  (1)  by  the  method  explained  in  Art.  230. 
So,  also,  the  term  containing  xz  may  always  be  made  to  dis-^ 
appear  by  a  new  transformation,  in  which  the  new  axis  of  y 
remains  unchanged ;  and  in  the  same  manner  the  term  con- 
taining yz  may  be  made  to  disappear.  Hence  equation  (1)  can 
always  be  transformed  into  an  equation  of  the  form 

Aaj'  +  B?/^  +  Cs''+Da?+E2/+F^4-G==0.  (2) 

If,  in  equation  (2),  neither  A,  B,  nor  C  is  zero,  we  may,  as  in 

Art.  229,  cause  the  terms  containing  the  first  powers  of  x,  y, 

and  z  to  disappear  by  changing  the  origin  of  the  co-ordinates, 

and  the  equation  will  be  reduced  to  the  form 

Laj'+My^+N^'+P^O.  (3) 

310.  Classification  of  the  surfaces  Tepresented  hy  the  equa- 
tion (3).  In  discussing  equation  (3)  we  must  suppose  each  of 
the  coefficients  to  be  either  plus  or  minus,  and  we  must  also 
consider  the  case  in  which  P  reduces  to  zero.    !N^ow  two  of  the 


234  ANALYTICAL    GEOMETRY. 

coefficients  L,  M,  and  N  must  always  have  tlie  same  sign  ;  we 
will  suppose  that  L  and  M  have  the  same  sign,  and  will  make 
these  signs  positive.  We  may  then  have  the  six  following 
cases : 

1.  When  ]Sr  is  plus  and  P  minus.     Equation  (3)  will  then 
take  the  form  Lx' + My'' + N^''  -  P  =  0. 

P         P  P 

If  we  divide  by  P,  and  put  a'=p  ^^=yry  ^-nd  0*=^,  we  shall 

«'   y"   ^"   -i 

have  -^+T^+-^=lj 

which  is  the  equation  of  the  surface  of  an  ellijpsoid  (Art.  307, 
eq.l). 

2.  When  l!^  is  plus  and  P  plus.     Equation  (3)  will  then  be- 
come Li»' + My' + InV + P = 0, 

in  which  all  the  terms  are  positive.  Hence  the  equation  can 
not  be  satisfied  for  real  values  of  the  variables,  and  therefore 
the  surface  becomes  imaginary. 

3.  When  N  is  plus  and  P  is  zero.     Equation  (3)  will  then 
become  Laj'+My'+Ns'^O, 

which  can  only  be  satisfied  by  the  values 

a?=0,        y=0,        s=0; 
and  hence  this  supposition  reduces  the  surface  to  2i. point^yiz., 
the  origin. 

4.  When  N  is  minus  and  P  is  minus.     Equation  (3)  will  then 

become  Laj'  +  M?/'-]Sr^''-P=0. 

P         P  P 

If  we  divide  by  P,  and  put  a^  —  j,  ^^"W  ^^^  ^'~T^'  ^®  ^^2X\. 

have  a''^b'''c'      ' 

which  represents  the  surface  of  an  hyjperholoid  of  one  sheet 

(Art.  307,  eq.  2). 

5.  When  !N"  is  minus  and  P  is  plus.    Equation  (3)  becomes 

L»'+M2/'~N2»+P=0. 
Substituting  as  before,  we  have 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE,  ETC.    235 

which  represents  the  surface  of  an  hyperholoid  of  two  sheets 
(Art.  307,  eq.  3). 

6.  When  N  is  minus  and  P  is  zero.     Equation  (3)  becomes 

which  by  substitution  becomes 

which  represents  the  surface  of  a  cone  having  an  elliptic  base 
(Art.  307,  eq.  4). 

311.  Particular  cases  of  the  general  equation.  If  both 
terms  containing  one  variable,  as  2,  are  wanting  from  eq.  (2), 
Art.  309,  that  is,  if  C  and  F  are  zero,  all  sections  of  the  surface 
perpendicular  to  the  axis  of  z  are  equal  to  each  other,  since  the 
equation  is  independent  of  z.  The  common  equation  of  these 
sections  is  Kx^ + B?/'^  +  D^ + Ey + G = 0, 

which  may  represent  either  of  the  conic  sections  (Art.  233). 
This  surface  is  called  a  cylindrical  surface^  and  may  be  de- 
scribed either — 

1.  By  the  above-named  conic  section  moving  always  parallel 
to  itself  and  along  a  right  line  parallel  to  the  axis  of  s,  or 

2.  By  a  straight  line  which  moves  along  the  conic  section, 
and  in  all  of  its  positions  is  parallel  to  the  axis  of  z. 

The  conic  section  is  called  the  hase  of  the  cylinder,  and  the 
cylinder  is  called  circular^  ellijptic^hyperholicy  ox  parabolic,  ac- 
cording to  the  nature  of  the  base. 

When  the  equation  A£c''4-By''-f  Daj-f-Ey+G=0  represents 
two  straight  lines  (Art.  233),  the  cylindrical  surface  becomes 
two  planes,  which  may  intersect  or  be  parallel,  or  may  coincide 
as  a  double  plane. 

When  two  of  the  three  coefficients  A,  B,  and  C  in  eq.  (2), 
Art.  309,  are  zero,  as  B  and  C,  one  of  the  corresponding  terms 
Ey  and  F^  may  be  made  to  disappear  by  a  transformation 
in  which  x  remains  unchanged,  but  the  axes  of  y  and  z  are 
changed  in  the  plane  yz^  and  the  resulting  equation  is  that  of 
a  cylinder,  as  above. 


236  ANALYTICAL   GEO^IETEY. 

312.  Ellijptic  and  JiyperholiG  jparaholoids.  The  only  re- 
maining case  of  eq.  (2),  Art.  309,  is  when  two  of  the  coeffi- 
cients, as  A  and  B,  are  finite,  and  the  third  is  zero.  The  first 
powers  of  x  and  y  can  then  be  made  to  disappear  by  changing 
the  origin  of  x  and  3/,  and  the  constant  term  may  be  made  to 
disappear  by  changing  the  origin  of  z.  The  equation  will  then 
become  Ax''  +  l^y''-\-Yz=0, 

which  may  be  written     -1 + Ti + - = 0. 

If  A  and  B  have  like  signs,  the  surface  is  that  of  an  ellvptic 
paraboloid  ;  if  A  and  B  have  unlike  signs,  every  cross  section 
perpendicular  to  the  axis  of  z  becomes  an  hyperbola,  and  the 
surface  is  called  an  hyperbolic  paraboloid. 

313.  How  an  elliptic  or  hyperbolic  paraboloid  7)iay  be  de- 
scribed. A  parabola  may  be  regarded  as  the  limiting  case  of 
an  ellipse,  one  vertex  of  which  is  fixed,  and  the  other  is  re- 
moved to  an  indefinitely  great  distance.  So,  also,  the  elliptic 
paraboloid  may  be  regarded  as  an  ellipsoid,  one  of  whose  axes 
has  been  indefinitely  increased,  while  one  vertex  of  that  axis 
remains  fixed. 

The  elliptic  paraboloid  may  be  regarded  as  described  by  one 
parabola  moving  upon  another.  Thus,  let  the  plane  of  one 
parabola  be  at  right  angles  to  the  plane  of  another;  let  the 
axes  of  the  two  parabolas  coincide,  and  the  concavities  be 
turned  in  the  sa7ne  direction.  Then,  if  one  of  the  parabolas 
move  so  as  to  be  always  parallel  to  itself  and  to  have  its  vertex 
upon  the  fixed  parabola,  the  surface  described  by  the  movable 
parabola  will  be  an  elliptic  paraboloid. 

But  if  the  concavities  of  the  two  parabolas  are  turned  in  op- 
posite directions,  the  corresponding  surface  thus  described  will 
be  2i\\  hyperbolic  paraboloid. 

314.  Section  of  a  surface  of  the  second  degree  by  a  plane. 
Every  intersection  of  a  plane  with  a  surface  of  the  second  de- 
gree is  either  a  straight  line  or  one  of  the  conic  sections. 


GENERAL   EQUATION   OF    THE    SECOND   DEGREE,  ETC.         237 

For  by  one  or  two  transformations  of  co-ordinates  like  those 
of  Art.  309  we  can  refer  the  surface  to  a  new  system  of  co-or- 
dinates, one  of  which,  as  ^,  will  be  parallel  to  the  given  inter- 
secting plane.  In  these  transformations  it  is  evident  that  the 
degree  of  the  equation  can  not  be  increased,  since  the  values 
substituted  for  x,  y,  and  z  are  always  of  the  first  degree.  If 
now  we  substitute  for  3  in  the  transformed  equation  the  dis- 
tance of  the  intersecting  plane  from  the  plane  xy,  we  shall 
have  an  equation  between  x  and  y,  which  is  the  equation  of 
the  intersection  of  the  plane  and  surface.  The  degree  of  this 
equation  does  not  exceed  the  second,  and  therefore  (Art.  233) 
the  curve  must  be  either  a  straight  line  or  a  conic  section. 

The  conic  section  may,  however,  in  special  cases,  break  up 
into  two  lines,  as  shown  in  Art.  233. 


APPENDIX. 


ON  THE  GRAPHICAL  REPRESENTATION  OF  NATURAL  LAWS. 
The  mutual  dependence  existing  between  any  two  or  more 
variable  quantities  may  be  exhibited  by  means  of  curve  lines. 
If,  for  example,  we  have  a  large  collection  of  meteorological 
observations  showing  the  temperature  at  any  place  for  each 
hour  of  the  day,  the  nature  of  the  relations  or  laws  expressed 
by  these  numbers  may  be  represented  by  curve  lines.  Such  a 
mode  of  representation  frequently  renders  these  laws  perfectly 
obvious,  and  sometimes  suggests  relations  which  might  easily 
have  been  overlooked  in  a  large  mass  of  figures  arranged  in 
tables.  There  is  a  great  variety  in  the  modes  by  which  this 
representation  may  be  effected.  The  following  are  some  of 
the  methods  most  frequently  employed : 

I.  Relations  of  two  variables  expressed  hy  rectangular  co- 
ordinates. If  on  a  horizontal  line  we  set  off  distances  propor- 
tional to  the  values  of  one  of  the  two  variables,  regarding  these 
as  abscissas,  and  from  the  several  points  of  division  erect  per- 
pendiculars whose  lengths  are  proportional  to  the  correspond- 
ing  values  of  the  other  variable,  and  then  draw  a  continuous 
curve  line  through  the  extremities  of  these  perpendiculars, 
this  curve  line  may  be  regarded  as  representing  the  relation 
between  the  two  variables.  The  cases  of  this  nature  most  fre- 
quently occurring  are  those  in  which  time  is  one  of  the  varia- 
bles, and  this  is  usually  laid  off  upon  the  axis  of  abscissas. 

Ex.  1.  Diicrnal  change  of  temperature.  Let  it  be  proposed 
to  construct  the  curve  which  represents  the  relation  between 
the  different  hours  of  the  day  and  the  corresponding  mean 
temperature  at  a  given  place.    The  following  table  shows  the 


240 


APPENDIX. 


mean  temperature  at  New  Haven  for  each  hour  of  the  day,  as 
deduced  from  a  long  series  of  observations : 


Hour. 

Temp. 

Hour. 

Temp. 

Hour. 

Temp. 

Hour. 

'Temp. 

Midnight 

45°.0 

6  A.M. 

43°.  1 

Noon. 

55°.  3 

6  P.M. 

52°.0 

1A.M. 

44  .3 

7     " 

44  .0 

1P.M. 

56  .1 

7     " 

60  .2 

2      " 

43  .6 

8      " 

46  .9 

2     " 

56  .5 

8     " 

48  .7 

3      " 

43  .1 

9      " 

49  .7 

3     " 

56  .3 

9     " 

47  .5 

4      " 

42  .7 

10      " 

52  .2 

4     " 

55  .4 

10     " 

46  .5 

5      " 

42  .6 

11      " 

54  .0 

5     " 

53  .9 

11     " 

45  .7 

--^ 

/ 

N 

/ 

V 

/ 

N, 

/ 

\ 

/ 

\ 

/ 

"^ 

--v^ 

In  order  to  represent  these  observations  by  a  curve  line,  we 

draw  upon  a  sheet  of  pa- 
per a  horizontal  line,  and 
divide  it  into  twenty-four 
equal  parts,  to  represent 
the  hours  of  the  day,  and 
through  these  points  of 
m't  2h.  4  G  8  10  uoou  2h.  4  6  8  10  m't  divisiou  WO  draw  a  system 
of  vertical  lines.  Upon  each  of  these  vertical  lines  we  set  off  a 
distance  proportional  to  the  height  of  the  thermometer  for  the 
corresponding  hour,  and  then  connect  all  the  points  thus  de- 
termined by  a  continuous  line.  The  curve  thus  formed  repre- 
sents the  mean  motion  of  the  thermometer  at  New  Haven  for 
the  different  hours  of  the  day,  and,  if  constructed  with  proper 
care,  and  upon  a  scale  of  suitable  size,  may  supply  the  place  of 
the  numbers  from  which  it  was  derived,  the  temperatures  being 
indicated  by  the-  numbers  on  the  left  of  the  diagram.  In  order 
to  avoid  confusion,  the  ordinates  in  the  diagram  have  only  been 
drawn  for  the  alternate  hours. 

We  readily  perceive  from  the  figure  that  on  each  day  there 
is  a  maximum  and  a  minimum  of  temperature,  the  maximum 
occurring  generally  about  two  hours  after  noon,  and  the  mini- 
mum about  an  hour  before  the  rising  of  the  sun.  We  see,  also, 
that  the  temperature  is  increasing  during  nine'hours  of  the  day, 
and  decreasing  during  the  remaining  fifteen  hours  of  the  day. 
This  curve  readily  shows  us  the  two  periods  of  the  day  when 
any  given  temperature  is  attained ;  as,  for  example,  a  tempera- 
ture of  50°,  52°,  etc.    It  also  shows,  not  only  the  mean  tem- 


APPENDIX. 


241 


perature  at  the  hours  of  observation,  but  also  for  any  time 
intermediate  between  these  hours;  as,  for  example,  for  each 
half  hour,  or  quarter  hour,  etc. 

Ex.  2.  Annual  change  of  temperature.  In  the  same  man- 
ner we  may  construct  the  curve  representing  the  connection 
between  the  different  months  of  the  year  and  the  correspond- 
ing temperature  at  a  given  place.  We  draw  a  horizontal  line, 
and  divide  it  into  twelve  equal  parts,  to  represent  the  months 
of  the  year,  and  through  these  points  of  division  draw  a  system 
of  vertical  lines,  upon  which  we  set  off  distances  proportional  to 
the  heights  of  the  thermometer  for  the  corresponding  months. 

The  following  table  shows  the  mean  temperature  of  New 
Haven  for  each  month  of  the  year,  as  deduced  from  a  long 
series  of  observations.  It  also  shows  the  average  maximum 
temperature  of  each  month,  and  the  average  minimum  tem- 
perature of  each  month : 


Mean 

Mtiximum 

Minimum 

Mean 

Maximum 

Minimum 

Temp. 

Temp. 

Temp. 

Temp. 

Temp. 

Temp. 

Jan.... 

26°,5 

49°.  6 

-1°.0 

July.. 

71°.  7 

90°.  8 

52°8 

Feb.... 

28  .1 

51  .3 

+  1  .0 

Aug.. 

70  .3 

88  ,6 

50  .0 

March. 

36  .1 

61  .6 

10  .7 

Sept.. 

62  .5 

83  .6 

37  .6 

April. . 

46  .8 

72  .6 

25  .4 

Oct... 

51  .1 

73  .2 

26  .7 

Mav... 

57  .3 

81  .3 

35  .5 

Nov.. 

40  .3 

63  .2 

17  .7 

June.. 

67  .0 

89  .3 

45  .9 

Dec. 

30  .4 

53  .1 

4  .5 

90° 


70 


GO 


In  the  annexed  fig- 
ure, the  middle  -^eurve 
line  shows  the  mean 
temperature  of  each 
month  of  the  year,  ac- 
cording to  the  pi^ced- 
ing  observationsjwhile 
the  upper  curve  shows 
the  average  maximum 
temperature,  and  the  20 
lower  curve  the  aver-  10 
age  minimum  temper- 
ature for  each  month 
of  the  year. 


/m 

AXIMl 

M  "^ 

N 

/ 

\ 

/ 

\, 

/ 

^^ 

__^ 

\ 

/ 

/ 

'MEA 

M      ^ 

s 

s 

/ 

/ 

\ 

\, 

/ 

/ 

^ 

^ 

/ 

^ 

X 

\, 

\ 

/ 

/ 

'mini 

.U«\ 

s 

/ 

/ 

\, 

\, 

/ 

/ 

\, 

\ 

/ 

/ 

\ 

^ 

/ 

\ 

\ 

/ 

/ 

\^ 

/ 

/^ 

\, 

-  / 

.^ 

50    *== 


30 


JNl      A      M 


J      A 


N      D 


242  APPENDIX. 

These  curves  inform  us  that  at  Kew  Haven  the.  warmest 
months  of  the  year  are  July  and  August,  and  the  maximum  for 
the  year  occurs  near  July  24th.  The  coldest  month  of  the 
year  is  January,  and  the  minimum  for  the  year  occurs  near 
Jan.  21st.  The  difference  between  the  minimum  and  the  max- 
imum for  each  month  is  greater  in  the  cold  months  than  in  the 
warm  months.  Various  other  particulars  respecting  the  con- 
nection, between  the  temperature  and  the  season  of  the  year 
are  also  exhibited  by  the  figure  more  palpably  than  by  a  col- 
umn of  numbers  in  a  table. 

The  same  mode  of  representation  may  be  employed  to  ex- 
hibit the  relation  between  the  height  of  the  barometer  and  the 
hour  of  the  day  or  the  season  of  the  year ;  also  for  the  amount 
of  vapor  in  the  atmosphere,  the  force  of  the  wind,  the  fall  of 
rain  or  snow,  the  prevalence  of  cloud  or  fog,  the  intensity  of 
atmospheric  electricity,  the  declination  or  dip  of  the  magnetic 
needle,  or  the  intensity  of  terrestrial  magnetism,  or,  indeed,  any 
natural  phenomenon  which  depends  on  the  course  of  the  sun. 

Ex.  3.  Display  of  November  meteors.  On  the  morning  of 
Nov.  14, 1866,  a  remarkable  display  of  meteors  was  witnessed 
in  England,  and  the  sudden  increase,  as  well  as  the  equally  sud- 
den decline  in  the  number  of  meteors,  is  exhibited  by  a  curve 
line  much  more  strikingly  than  could  be  done  by  a  simple  nu- 
merical statement.  For  this  purpose  we  draw  a  horizontal 
line,  and  divide  it  into  equal  parts,  to  represent  the  hours  of 
observation,  and  througli  the  points  of  division  we  draw  a  sys- 
tem of  vertical  lines.  On  these  vertical  lines  we  set  off  dis- 
tances proportional  to  the  number  of  meteors  counted  each 
minute,  and  through  the  points  thus  determined  we  draw  a 
continuous  curve  line.  The  numbers  on  the  left  margin  of 
the  figure  on  the  opposite  page  indicate  the  number  of  meteoi*s 
visible  each  minute.  From  the  diagram  we  perceive  that  be- 
fore midnight  the  number  of  meteors  did  not  exceed  5  per 
minute,  but  soon  after  midnight  the  number  rapidly  increased, 
and  at  Ih.  20m.  exceeded  120  per  minute ;  by  2  A.M.  it  had  de- 
clined to  40  per  minute,  and  by  3  A.M.  to  10  per  minute. 


ArpENDix.  243 

laPM  II  MIDNIGHT       IM. 2  34  5 


/20. 


100 


80 


60 


A 

r 

,     \ 

\ 

l\l 

[ 

1 

/ 

\J 

/ 

V 

/ 

^-^v 

J 

"- ''V 

-^ 

A  similar  mode  of  representation  may  be  advantageously 
employed  to  exhibit  the  results  of  a  large  mass  of  observations, 
even  though  we  have  no  previous  knowledge  of  the  IfiAvs  which 
govern  their  changes.  "We  may  thus  exhibit  the  influence  of 
the  day  or  the  season  of  the  year  upon  mortality ;  we  may  ex- 
hibit the  average  number  of  deaths  at  different  ages ;  or  we 
may  exhibit  the  fluctuations  in  the  price  of  any  article  of  mer- 
chandise, as  wheat,  cotton,  gold,  etc. 

.  Ex.  4.  Annual  change  in  the  depth  of  rivers.  Tlie  depth  of 
the  water  in  the  Mississippi  Kiver  fluctuates  greatly  with  the 
season  of  the  year.  During  the  early  part  of  autumn  the  water 
is  usually  lowest,  and  it  is  highest  some  time  in  the  spring  or 
the  early  part  of  summer.  The  figure  on  the  following  page 
shows  the  average  result  of  twenty-three  years  of  observations 
on  the  river  at  Katchez,  Miss.  The  months  are  shown  at  the 
bottom  of  the  figure,  while  the  depth  of  water  is  indicated  by 
the  numbers  on  the  left  margin. 

"We  see  from  this  figure  that  the  water  is  usually  lowest  in 
October,  when  its  depth  is  only  12.5  feet.  From  this  time  the 
water  rises  pretty  steadily  to  the  first  of  May,  when  the  depth 
amounts  to  48.3  feet,  from  which  time  it  declines  pretty  steadily 


244 


APPENDIX. 


till  the  following  Oc- 
tober. There  are, 
however,  two  small- 
er maxima  which  are 
well  marked,viz.,  one 
about  the  1st  of  Feb- 
ruary, and  the  other 
about  the  middle  of 
June.  These  great 
fluctuations  of  the 
Mississippi  are  due 
not  so  much  to  an 
excess  of  rain  near 
the  time  of  maxi- 
mum height  as  to 
the  melting  of  the 
snow  accumulated 
upon  the  numerous 
tributaries  of  this 
river. 

Ex.  5.  Velocity   of 
the  current  of  a  riv- 


It  has  been  found  by  experiment  that  the  velocity  of  the  cur- 
rent in  rivers  varies  sensibly  with  the  depth.  This  may  be 
shown  by  means  of  floats  immersed  to  different  depths  in  the 
water.  The  following  is  one  mode  of  performing  the  experi- 
ment :  A  keg  15  inches  high  and  10  in  diameter,  without  top 
or  bottom,  is  ballasted  with  lead  so  as  to  sink  and  remain  up- 
right in  the  water ;  the  keg  is  attached  by  a  small  cord  to  a 
mass  of  cork  8  inches  square  and  3  inches  thick,  and  a  small 
flag  is  supported  by  the  cork,  in  order  that  it  may  be  more 
readily  observed  at  a  distance.  By  varying  the  length  of  the 
cord,  the  keg  may  be  made  to  sink  to  any  required  depth,  and 
its  size  is  so  much  greater  than  that  of  the  surface-float  that 
the  latter  does  not  sensibly  affect  the  rate  of  movement. 


60 
48 

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38 
S6 
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18 
16 

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11 

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\y 

n™. 

Jau. 

Feb. 

Mar. 

Apr. 

May 

Jun. 

Jul. 

Aug. 

Sep. 

Oct. 

Not; 

APPENDIX. 


245 


Surface  3.5 


0.1 


8.7 


S.S 


0.3 


0.5 


The  apparatus  being  placed  in  the  water,  its  rate  of  motion 
is  determined  by  observers  stationed  on  the  bank  of  the  river 
at  known  distances  from  each  other,  and  watching  the  progress 
of  the  float  by  means  of  theodolites. 

The  curve  line  on  the  annexed  figure  shows  the  result  of 
experiments  made  on  the  cur- 
rent of  the  Mississippi  near  Kew 
Orleans.  The  numbers  on  the 
left  margin  show  the  depth  of 
the  keg,  expressed  in  tenths  of 
the  entire  depth  of  the  river,  the 
mean  depth  of  the  w^ater  being 
SQ  feet.  The  numbers  at  the 
top  of  the  figure  show  the  ve- 
locit}'^  of  the  current,  expressed  ^-^ 
in  miles  and  tenths  of  a  mile 
per  hour. 

We  see  from  the  figure  that 
the  velocity  at  the  surface  is 
3.74  miles  per  hour ;  the  velocity  increases  as  we  descend,  until 
we  reach  a  depth  about  one  third  that  of  the  river,  where  the 
velocity  amounts  to  3.84  miles  per  hour,  while  below  this  depth 
the  velocity  diminishes,  and  at  the  bottom  of  the  river  is  re- 
duced to  3.47  miles  per  hour. 

Ex.  6.  Average  duration  ofhuonan  life.  The  average  du- 
ration of  life  may  be  deduced  from  tables  which  show  the 
number  of  deaths  which  occur  each  year  out  of  a  given  num- 
ber of  individuals.  If  there  were  a  million  of  births  in  the 
year  1770,  and  we  had  a  record  of  the  number  of  deaths  out 
of  this  company  for  each  year  to  the  present  time,  we  could 
construct  a  table  showing  the  average  duration  of  life  for  each 
age.  The  average  duration  of  life  for  a  person  of  a  certain 
age  is  understood  to  be  the  average  number  of  years  which 
the  survivors  of  that  age  should  live.  The  duration  of  life  is 
different  in  different  countries.  The  curve  line  in  the  follow- 
ing figure  shows  the  average  duration  of  life  as  deduced  from 


Bottom 


246 


APPENDIX. 


CO 
45 
40 

/ 

^ 

k 

/ 

\ 

s 

\ 

\ 

10 

\ 

s. 

25 
20 
15 

10 

\ 

[\ 

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^ 

— 

£0  35  40  45   60  65  CO   65  70  75 


85  i^O  95  100 


observations  made  at  Carlisle,  Eng.  The  numbers  at  the  bot- 
tom of  the  fierm-e  show  the  ao;e  of  the  individual  from  0  to  100 
years,  and  the  numbers  on  the  left  margin  show  the  average 
duration  of  life.  This  average  duration  of  the  life  of  individ- 
uals after  any  specified  age  is  called  the  expectatio?i  of  life. 

We  see  from  the  figure  that  the  average  duration  of  life 
for  an  infant  just  born  is  38  years.  If  the  child  survives,  its 
expectation  of  life  increases  for  a  few  years,  and  attains  its 
maximum  at  the  age  of  5,  when  the  average  duration  of  life  is 
51  years.  After  this  age  the  average  duration  of  life  dimin- 
ishes steadily  and  pretty  uniformly  until  death.  At  the  age  of 
25  the  average  duration  of  life  is  38  years,  at  50  it  is  21  years, 
at  75  it  is  Y  years,  and  at  100  it  is  2  years. 


II.  Belations  of  several  variables  defending  upon  a  com- 
mon variable.  When  we  have  several  variable  elements  de»- 
pending  upon  a  common  variable,  we  may  graduate  a  horizon- 
tal line  to  represent  successive  values  of  the  common  variable, 
and  then  construct  a  number  of  curve  lines  to  represent  the 
changes  in  each  of  the  other  variables.  A  comparison  of  the 
different  curves  will  show  not  only  the  relation  of  each  variable 
to  the  common  variable,  but  also  the  mutual  relation  of  the 
several  variables. 


APPENDIX. 


247 


Apr.May  Jun.  J 
70i 


Aug.  Sep.  Oct.  yov.  Dee.  Jan.  Teb.Mar.Apr. 


Ex.  1.  Tempemture  helow  the  eartKs  surface.  Suppose  we 
wish  to  discover  how  the  diurnal  and  annual  changes  of  tem- 
perature are  modified  by  depth  below  the  surface  of  the  earth. 
For  this  purpose  we  require  observations  of  temperature  made 
at  different  depths  below  the  earth's  surface,  and  continued  at 
least  throughout  an  entire  year.  Such  observations  have  been 
made  at  several  places  in  Europe.  Thermometers  with  very 
long  stems  have  been  buried  at  depths  of  24, 12, 6,  and  3  French 
feet,  and  1  inch,  and  the  observations  have  been  continued  for 
many  years.  The  annexed  figure  presents  a  summary  of  such 
observations  contin- 
ued for  14  years  at 
Greenwich,  the 
months  being  given  ^ 
at  the  top  of  the  fig-  ^^ 
Tire,  and  tlie  temper-  ^ 
atures  on  the  left  ^ 
margin.  ^^ 

We  perceive  from  ^ 
the  figure  that  at  a  53 
depth  of  about  6  feet  eo 
the  annual  range  of  43 
temperature  is  only  4c 
about  half  what  it  44 
is  at  the  surface ;  at  42 
the  depth  of  12  feet  *» 
the  annual  range  of  temperature  is  less  than  one  third,  and  at 
the  depth  of  24  feet  it  is  only  one  ninth  what  it  is  at  the  sur- 
face. We  also  perceive  that  the  highest  temperature  of  the 
year  occurs  later  and  later  as  we  descend  below  the  surface  of 
the  earth.  At  tlie  depth  of  12  feet  the  maximum  temperature 
of  the  year  occurs  about  the  last  of  September,  and  the  mini- 
mum about  the  last  of  March,  while  at  the  depth  of  24  feet  the 
maximum  occurs  about  the  first  of  December,  and  the  minimum 
about  the  first  of  June. 

Ex.  2.  Declination  of  the  magnetic  needle  and  the  solar 


248 


APPENDIX. 


spots.  The  surface  of  the  sun  often  exhibits  black  spots  of 
irregular  form  and  variable  size.  The  number  of  these  spots 
varies  greatly  in  different  years ;  sometimes  the  sun  is  entirely 
free  from  them,  and  continues  thus  for  months  together,  while 
some  years  the  sun  is  never  seen  entirely  free  from  spots.  The 
curve  in  the  lower  part  of  the  annexed  figure  presents  a  sum- 


1810 


: :t :::::::_: 

magn|:tic        \            /\|otedle  > 

:             fc'f  v"4-t--:X---4 

t\l   \  -lX-X\-^4 

^\'/\--X  1  __L/ \_-l 

'T^/^     x/      ^       V        %      v^ 

100                                       ^            2V           T 

CO                                                                        H        A                           \       T'     3           i 

t                            ^\    X           -,       \-            ^      I 

^            7   I    t    _!_      i         X-   -.         V   t- 

'°    it    T^-^         ^    3    t      \    4-         X   ^         XX 

t  X     J-       ^t     ^^t       X  t\     ^ 

\      ^       \      L         V          ^^         ^   h         ^ 

1870 


mary  of  observations  of  the  spots  for  a  period  of  64  years,  the 
dates  being  given  at  the  bottom  of  the  figure, while  the  fre- 
quency of  the  spots  is  exhibited  on  the  left  margin  by  a  scale 
of  numbers  extending  from  0  to  100.  We  readily  perceive 
that  the  spots  are  subject  to  a  certain  periodicity,  the  number 
of  the  spots  increasing  for  5  or  6  years,  and  then  decreasing 
for  several  years,  showing  alternate  maxima  and  minima.  The 
maxima  occurred  in  1817, 1830, 1837, 1848,  and  1860,  while 
the  minima  occurred  in  1810, 1823, 1833, 1843, 1856,  and  1867. 
A  magnetic  needle,  when  freely  suspended  and  carefully  ob- 
served from  hour  to  hour,  exhibits  a  small  daily  oscillation  va- 
rying from  5'  to  15'.  The  extent  of  this  oscillation  varies  with 
the  season  of  the  year,  and  the  mean  annual  range  varies  from 
one  year  to  another.  The  curve  in  the  upper  part  of  the  above 
figure  shows  the  results  of  observations  of  the  magnetic  needle 


APPENDIX. 


249 


made  in  Europe  for  a  period  of  64  years,  the  dates  being  shown 
at  the  bottom  of  the  figure,  and  the  mean  daily  average  of  the 
needle  being  shown  by  numbers  on  the  left  margin,  which  rep- 
resent minutes  of  arc. 

We  see  from  the  figure  that  the  range  of  the  needle,  which 
was  only  6'  in  1810,  had  increased  to  8^  in  1818,  had  decreased 
again  to  about  6'  in  1824,  and  increased  to  10'  in  1829,  etc.  In 
other  words,  the  annual  range  of  the  magnetic  needle  shows 
alternate  maxima  and  minima,  and  the  times  of  these  maxima 
correspond  remarkably  with  the  maxima  of  the  solar  spots, 
suggesting  the  idea  that  the  two  phenomena  are  dependent 
upon  a  common  cause.  Such  a  mode  of  representation  by 
curve  hues  is  well  calculated  to  show  the  connection  between 
two  different  classes  of  phenomena. 


III.  delations  of  two  variables  expressed  hy,  jpolar  co-ordi- 
nates. The  relations  between  two  variable  elements  may  be 
expressed  by  means  of  polar  co-ordinates,  and  this  method  is 
generally  to  be  preferred  when  one  of  the  variables  denotes 
direction  ;  for  example,  if  one  of  the  variables  is  the  direction 
of  the  wind,  and  the  other  variable  is  the  corresponding  mean 
height  of  the  barometer,  or  thermometer,  or  hygrometer.  For 
example,  suppose  we  wish  to  show  the  dependence  of  the  tem- 
perature of  the  air  upon  the  direction  of  the  wind. 

Ex.  1.  Influence  of  the  wind  on  temperature.  From  a  com- 
parison of  several  years  of  observations,  it  has  been  found  that 
at  New  Haven  the  temperature  of  the  air  during  the  prevalence 
of  winds  from  the  eight  principal  points  of  the  compass  diffei'S 
from  the  mean  temperature  of  the  year  by  the  quantities  shown 
in  the  annexed  table : 


Wind. 

Temperature. 

Wind. 

Temperature. 

North 

Northeast 

East 

Southeast 

—  2°.  7 

-0  .*6 
+0  .5 
+  1  .2 

South 

Southwest.... 

West 

Northwest 

+3°.2 
+4  .0 
-1  .1 
-4  .5 

In  order  to  represent  these  results  by  a  curve  line,  we  draw 

L2 


250 


APPENDIX. 


eiglit  radii  inclined  to  each  oth- 
er in  angles  of  45°,  to  represent 
the  directions  of  the  wind.  With 
tlie  point  A  as  a  centre,  we  draw 
a  series  of  equidistant  circum> 
f erences,  to  represent  differences 
of  temperature,  and  then,  having 
selected  one  of  these  to  represent 
the  mean  temperature  of  New 
Haven,  w^e  set  off  upon  the  eight 
radii  distances  proportional  to 
the  numbers  in  the  preceding  table.  When  the  numbers  are 
negative,  we  set  them  off  towards  the  centre  of  the  circle ; 
when  they  are  positive,  we  set  them  o^from  the  centre.  The 
curve  line  passing  through  the  eight  points  thus  determined 
shows  the  influence  of  the  wind's  direction  upon  the  tempera^ 
ture  of  the  air.  We  perceive  that  the  highest  temperature  ac- 
companies a  wind  from  S.  33°  W.,  and  the  lowest  temperature 
corresponds  to  a  wind  from  the  point  N.  40°  W.,  the  mean  dif- 
ference in  the  temperature  of  these  two  winds  being  8°.7. 

Ex.  2.  Direction  of  the  prevalent  wind.    The  prevalent  wind 
at  any  station  may  be  graphically  represented  by  means  of  polar 

co-ordinates.  Suppose  we  have  a 
long  series  of  observations  of  the 
wind  from  which  we  deduce  the 
number  of  times  the  wind  was  ob- 
served to  blow  from  the  north 
point;  also  the  number  of  times 
it  blew  from  the  northwest,  the 
number  of  times  from  the  west, 
and  so  on,  for  8  or  16  points  of  the 
compass.  We  draw  two  lines  at 
right  angles  to  each  other  to  rep- 
resent the  cardinal  points,  and  also 
other  lines  to  represent  the  interme- 
diate directions.     From  the  point 


APPENDIX.  251 

of  intersection  we  set  off  on  these  lines  distances  correspond- 
ing to  the  relative  frequency  of  the  winds  from  these  different 
points  of  the  compass.  The  curve  line  passing  through  the 
points  thus  determined  may  be  regarded  as  showing  the  prev- 
alent wind  for  that  station. 

The  preceding  -Qgure  shows  the  results  of  observations  made 
during  the  month  of  January  for  several  years  at  Wallingford, 
near  New  Haven.  We  see  that  tlie  prevalent  wind  is  almost 
exactly  from  the  north,  but  that  winds  from  the  S.S.W.  are  also 
of  frequent  occurrence. 

Thi^  mode  of  representation  is  valuable  when  we  wish  to 
exhibit  the  peculiarities  of  a  large  number  of  stations.  The 
eye  is  thus  able  at  a  glance  to  detect  characteristic  peculiarities 
which  might  be  easily  overlooked  in  a  large  collection  of  nu- 
merical results. 

Ex.  3.  Diurnal  change  in  the  direction  of  the  wind.  An- 
other mode  of  representation,  bearing  some  resemblance  to  the 
preceding,  may  be  advantageously  employed  to  denote  the  con- 
nection between  the  hour  of  the  day  and  the  corresponding 
direction  of  the  wind.  Suppose,  from  a  long  series  of  observa- 
tions, we  have  determined  the  mean  direction  of  the  wind  for 
each  hour  of  the  day.  Having  drawn  two  lines  at  right  angles 
to  each  other  to  represent  the  cardinal  points  of  the  compass, 
we  begin  with  the  observation  for  the  first  hour,  and  draw  a 
line  of  any  convenient  length  to  represent  the  wind's  direction 
at  that  hour ;  from  the  extremity  of  this  line  we  draw  a  line 
of  .the  same  length  as  before,  to  represent  the  wind's  direction 
at  the  second  hour,  and  in  the  same  manner  we  set  off  the  di- 
rections of  the  wind  for  cacli  of  the  twenty-four  hours.  We 
thus  construct  a  broken  line,  which  may  be  regarded  as  repre- 
senting the  average  progress  of  a  particle  of  air  for  each  hour 
of  the  day,  supposing  tlie  wind's  velocity  to  have  been  the  same 
at  all  hours ;  or,  if  we  have  observations  showing  the  wind's 
velocity  for  each  hour,  we  may  make  the  portions  of  the  curve 
which  represent  the  wind's  direction  for  the  different  hours  rep- 
resent not  only  its  direction,  but,  at  the  same  time,  its  velocity. 


252 


APPENDIX. 


The  annexed  figure  shows  the  mean  di- 
rection of  the  wind  at  New  Haven  for  the 
different  honrs  of  the  day  during  the  month 
of  August.  We  perceive  that  early  in  the 
morning  the  average  direction  of  the  w^ind 
for  this  month  is  from  the  north,  while  dur- 
inn;  the  r.ftcrnooii  its  averao:e  direction  is 
from  the  south,  and  about  10  A.M.  the  wind 
veers  from  K  to  S.,  going  round  by  the  east. 
This  diurnal  change  in  the  wind's  direction 
constitutes  what  is  commonly  known  by 
the  name  of  a  *'  land  and  sea  breeze." 
The  change  in  the  wind's  direction  for  the  other  months  of 
the  year  may  be  represented  in  a  similar  manner. 


lY.  Contour  lines  and  geograjpJiical  distribution.  If  it  is 
required  to  represent  upon  a  map  the  undulations  in  the  surface 
of  a  tract  of  land,  we  suppose  the  surface  of  the  ground  to  be 
intersected  by  a  number  of  horizontal  planes  at  equal  distances 
from  each  other,  and  w^e  delineate  on  paper  the  curve  lines  in 
which  these  planes  intersect  the  surface. 

Ex.  1.  Survey  of  an  undulating  surface.  This  method  will 
be  understood  from  the  annexed  figure,  which  represents  a 


APPENDIX. 


25a 


tract  of  broken  ground  divided  by  a  stream,  EF.  The  ground 
is  supposed  to  be  intersected  by  a  horizontal  plane  four  feet 
above  F,  the  lowest  point  of  the  tract,  and  this  plane  intersects 
the  surface  of  the  ground  in  the  undulating  lines  marked  4,  one 
on  each  side  of  the  stream.  A  second  horizontal  plane  is  sup- 
posed to  be  drawn  eight  feet  above  F,  and  this  intersects  the 
surface  of  the  ground  in  the  lines  marked  8.  In  like  manner, 
other  horizontal  planes  are  drawn  at  distances  of  12, 16,  etc., 
feet  above  the  point  F.  The  projection  of  these  lines  upon 
paper  shows  at  a  glance  the  outline  of  the  tract. 

Ex.  2.  Depth  of  water  in  a  harbor.  If  we  have  soundings 
showing  the  depth  of  water  at  numerous  points  of  a  harbor, 
the  results  may  be  delineated  on  paper  in  a  similar  manner. 
We  draw  a  curve  line  joining  all  those  points  where  the  depth 
of  water  is  the  same — for  example,  10  feet.  We  draw  another 
line  connecting  all  those  points  where  the  depth  of  water  is  20 
feet ;  also  other  lines  for  30  feet,  40  feet,  etc. 

The  accompanying  figure  represents  a  portion  of  New  York 
Harbor,  and  the  dotted  lines  show  depths  of  20, 40,  and  60  feet 
of  water.     We  see  that  along  the  channel  of  the  North  River 


254  APPENDIX. 

there  is  every  wliei'e  a  depth  of  at  least  40  feet,  but  in  passing 
from  the  North  River  to  East  River  there  are  obstructions 
where  the  depth  of  water  is  only  20  feet. 

A  similar  principle  is  now  very  extensively  employed  to  rep- 
resent almost  every  variety  of  variable  quantity  depending 
upon  geographical  position.  In  many  cases  the  representation 
is  greatly  assisted  by  variations  in  the  depth  of  shading,  or  by 
varieties  of  color,  etc.  The  following  examples  will  afford 
some  idea  of  this  method. 

Ex.  3.  Lines  of  equal  mean  temperature.  We  draw  upon  a 
map  of  the  earth  a  curve  line  connecting  all  those  places  whose 
mean  temperature  is  the  same — for  example,  80°.  As  it  may 
happen  that  we  have  no  station  whose  observed  temperature  is 
exactly  80°,  we  select  two  adjacent  stations,  at  one  of  which 
the  temperature  is  a  little  less  than  80°,  and  at  the  other  a 
little  greater;  we  then  divide  the  interval  between  them  in 
the  same  ratio  as  the  differences  between  the  observed  temper- 
atures and  80°.  The  point  thus  determined  we  call  a  point  of 
80°  temperature.  In  the  same  manner  we  determine  as  many 
points  of  this  line  as  practicable.  Kext  we  draw  a  line  con- 
necting all  those  places  whose  mean  temperature  is  70°,  60°, 
50°,  etc.  The  figure  on  the  opposite  page  exhibits  such  a  sys- 
tem of  lines  for  nearly  the  entire  globe.  Maps  of  this  kind, 
when  carefully  constructed,  give  a  much  clearer  idea  of  the 
distribution  of  heat  on  the  earth's  surface  than  can  be  done  by 
any  system  of  numbers  arranged  in  tables. 

In  like  manner  we  may  draw  lines  representing  the  mean 
temperature  of  different  places  for  any  month  of  the  year,  or 
we  may  draw  lines  to  represent  the  temperatures  observed  for 
any  given  day  and  hour,  thus  enabling  us  to  study  the  actual 
distribution  of  temperature  at  any  instant  of  time. 

Ex.  4.  Lines  of  equal  atmospheric  pressure.  We  may  draw 
upon  a  map  of  the  earth  a  curve  line  connecting  all  those  places 
where  the  mean  pressure  of  the  air,  as  shown  by  a  barometer, 
is  the  same — for  example,  30  inches.  We  may  also  draw  lines 
connecting  those  places  wliere  the  mean  pressure  is  29.9  incli- 


APPENDIX. 


255 


es,  also  29.8  inches,  etc. ;  or  we  may  draw  lines  connecting  all 
those  places  where  the  pressure  is  the  same  at  any  given  day 
and  hour,  thus  enabling  us  readily  to  follow  the  daily  fluctua- 


256  APPENDIX. 

tions  attending  the  progress  of  storms  over  tlie  surface  of  the 
earth. 

The  annexed  figure  shows  the  state  of  the  barometer  and 
the  direction  of  the  wind  as  observed  near  the  centre  of  a  vio- 
lent storm  which  prevailed  in  the  neighborhood  of  'New  York 
February  16, 1842.     The  small  oval  line  shows  the  area  wdthin 


^\I?uc^este;y  ^ 


\ 


^.70ineh 


which  the  barometer  sunk  eight  tenths  of  an  inch  below  the 
mean,  and  the  larger  oval  shows  the  area  within  which  the 
barometer  was  depressed  seven  tenths  of  an  inch.  The  long 
arrow  represents  the  direction  in  which  the  storm  advanced, 
while  the  short  arrows  show  the  observed  direction  of  the  wind 
at  nearly  forty  different  stations. 

Ex.  5.  Lines  of  equal  magnetic  declination,  di^,  etc.  We 
may  draw  upon  a  map  of  the  earth  curve  lines  connecting  all 
those  places  where  the  declination  of  the  magnetic  needle  is 
the  same,  or  where  the  dip  of  the  magnetic  needle  is  the  same, 
or  the  earth's  magnetic  intensity  is  the  same.     Such  lines  give 


APPENDIX.  257 

a  far  more  distinct  idea  of  the  distribution  of  magnetism  over 
the  earth's  surface  than  could  be  furnished  by  any  amount  of 
numerical  results  exhibited  in  a  tabular  form. 

The  annexed  figure  shows  the  lines  of  equal  magnetic  decll- 
nation  for  a  portion  of  the  United  States  for  the  year  1850. 


We  perceive  that  the  line  of  no  declination  passed  through  the 
centre  of  Lake  Erie,  and  met  the  Atlantic  near  the  middle  of 
the  coast  of  North  Carolina.  The  line  of  10  degrees  west  dec- 
lination passed  near  Montreal,  and  the  line  of  8  degrees  east 
declination  passed  near  St.  Louis.     Tliese  lines  show  a  small 


258  APPENDIX. 

motion  from  year  to  year,  and  at  present  tliey  all  have  a  posi- 
tion westward  of  the  positions  represented  on  the  map. 

The  map  also  shows  the  line  of  65°  magnetic  dip,  of  70°,  and 
of  75°  dip. 

Ex.  6.  How  the  principal  jphenotnena  of  a  storm  Tnay  he 
represented.  "Winter  storms  in  the  United  States  are  of  great 
extent,  sometimes  exceeding  1000  miles  in  diameter.  In  order 
to  represent  the  phenomena  of  such  a  storm,  we  require  some 
suitable  means  of  designating  the  area  upon  which  rain  or 
snow  is  falling ;  we  wish  to  denote  the  region  around  the  mar- 
gin of  the  storm  where  clouds  prevail  without  rain ;  and  we 
wish  to  represent  the  region  of  clear  sky  which  encircles  the 
storm  on  every  side.  We  wish  also  to  represent  the  depression 
of  the  barometer  within  the  storm  area ;  also  the  state  of  the 
thermometer  and  the  direction  of  the  wind  for  each  station  of 
observation.  The  mode  of  accomplishing  some  of  these  ob- 
jects will  be  understood  from  the  figure  on  the  opposite  page, 
which  represents  the  principal  phenomena  of  a  violent  storm 
which  was  experienced  in  the  United  States  December  20, 
1836.     The  map  represents  the  phenomena  for  8  P.M. 

The  deeply  shaded  portion  in  the  middle  of  the  figure  rep- 
resents the  area  where  rain  or  snow  was  falling;  the  lighter 
shade  on  the  east  and  west  margins  of  the  rain  represents  the 
region  where  clouds  prevailed  without  rain.  Throughout  the 
remaining  portion  of  the  United  States,  as  far  as  the  map  ex- 
tends, clear  sky  prevailed. 

The  dotted  curve  lines  represent  the  state  of  the  barometer. 
The  inner  curve  shows  the  area  where  the  barometer  was  de- 
pressed four  tenths  of  an  inch  below  the  mean ;  the  next  curve 
shows  where  the  barometer  was  two  tenths  of  an  inch  below 
the  mean;  the  next  curve  shows  the  barometer  at  its  mean 
height ;  while  farther  eastward  the  barometer  stood  two  tenths 
of  an  inch  and  four  tenths  of  an  inch  above  the  mean. 

The  arrows  show  the  directions  of  the  wind  as  observed  at  a 
large  number  of  stations. 

A  similar  map,  constructed  for  8  A.M.,  December  21,  would 


APPENDIX. 


259 


05  Rn  n"^  sr 


METEOROLOGICAL   CHART 


^^^         For  8  P.M. Dec.  20. 1836 


show  not  only  that  the  storm  had  traveled  eastward,  but  tliat 
important  changes  had  taken  place  within  the  storm  area. 

This  mode  of  representing  the  phenomena  of  a  storm  not 
merely  compresses  a  vast  amount  of  information  within  a  small 
space,  but  it  constitutes  a  powerful  instrument  of  research,  as 
it  indicates  a  connection  between  the  different  classes  of  obser- 
vations which  might  entirely  escape  notice  if  the  comparisons 
were  limited  to  a  collection  of  observations  arranged  in  a  tab- 
ular form. 


V.  Belations  of  three  independent  variables.  Since  two  co- 
ordinates are  required  to  determine  the  position  of  a  point  on 
a  plane,  every  point  of  a  plane  may  be  considered  as  corre- 
sponding to  the  known  values  of  two  of  the  variable  elements. 
Take  now  three  correspondinij  values  of  the  three  elements; 


260  APPENDIX. 

set  off  two  of  them  as  abscissa  and  ordinate  on  the  given  plane, 
and  at  the  point  thus  determined  erect  a  perpendicular  whose 
length  is  proportional  to  the  corresponding  value  of  the  third 
element.  Proceed  in  the  same  manner  with  every  three  cor- 
responding values  of  the  three  variables.  The  extre^lities  of 
all  these  perpendiculars  will  be  situated  upon  a  curved  surface 
which  represents  the  law  connecting  the  three  variable  ele- 
ments. Suppose  now  a  system  of  equidistant  planes  to  be 
drawn  parallel  to  the  plane  first  assumed ;  these  planes  will  in- 
tersect the  curved  surface  in  curve  lines  w^hose  form  will  indi- 
cate the  undulations  of  that  surface.  Let  these  curves  be  now 
projected  on  the  plane  first  assumed,  and  we  shall  have  on  a 
single  plane  a  system  of  curve  lines  which  give  a  precise  idea 
of  the  changes  of  the  third  variable  corresponding  to  any  given 
change  of  the  other  two  variables. 

Ex.  Temperature  at  any  hour  and  for  any  month.  Let  it 
be  required  to  represent  to  the  eye,  by  means  of  curve  lines, 
the  mean  temperature  of  a  given  place  for  any  hour  of  the  day 
or  any  month  of  the  year.  We  mark  off  on  the  axis  of  abscis- 
sas equal  divisions  to  represent  the  months  of  the  year,  and  on 
the  axis  of  ordinates  we  set  off,  in  like  manner,  twenty-four 
equal  divisions  to  represent  the  hours  of-  the  day,  and  through 
these  points  of  division  we  draw  lines  parallel  to  the  co-ordinate 
axes.  We  are  supposed  to  have  a  table,  derived  from  observa- 
tion, which  shows  us  the  mean  temperature  of  the  given  place 
for  each  hour  and  each  month  of  tlie  year.  We  now  select 
any  temperature — for  example,  32° — and  find  the  two  hours  of 
each  month  at  which  that  temperature  occurs.  At  the  inter- 
section of  the  abscissa  and  ordinate  corresponding  to  the  given 
month  and  hour  we  place  a  point,  and  we  do  the  same  for  each 
of  the  dates  where  the  given  temperature  occurs.  We  join  all 
these  points  by  a  continuous  curve  line,  and  we  have  a  re23re- 
sentation  of  the  curve  of  32°.  In  like  manner  we  draw  the 
curve  of  30°,  of  28°,  etc.,  through  the  entire  range  of  the  ob- 
servations. The  figure  on  the  opposite  page  shows  the  results 
of  a  Ions:  series  of  observations  at  New  Haven. 


APPENDIX. 


261 


October     November  December    January      February      March  April 


May 


Such  a  figure  shows  at  a  glance  the  mean  temperature  cor- 
responding to  any  hour  of  either  month  of  the  year.  If,  for 
example,  we  desire  to  know  the  mean  temperature  of  the  month 
of  January  at  6  A.M.,  we  find  6  A.M.  on  the  left  margin  of  the 
table,  and  follow  along  the  corresponding  horizontal  line  until 
we  reach  the  middle  of  the  month  of  January.  The  point 
falls  nearly  on  the  curve  of  22°,  which  is  therefore  the  temper- 
ature sought.  In  like  manner  we  may  find  the  temperature 
corresponding  to  any  hour  of  any  month  of  the  year. 

The  same  figure  shows  the  season  of  the  year  and  the 
hour  of  the  day  when  the  lowest  temperature  occurs.  It  also 
shows,  for  any  season  of  the  year,  the  two  hours  which  have 
the  same  temperature ;  also,  for  any  hour  of  the  day,  the  two 
seasons  of  the  year  which  have  the  same  temperature.  It  also 
shows  when  the  temperature  changes  most  slowly,  and  when 
it  changes  most  rapidly. 

In  a  similar  manner  we  may  construct  a  system  of  curve 
lines  representing  the  relation  between  any  three  independent 
variables. 


THE    END. 


\.  I 


Harper's  Catalogue. 


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